semi lagrangian and semi implicit discontinuous galerkin methods for atmospheric ... ·...

POLITECNICO DI MILANODottorato di Ricerca inIngegneria Matematica

– XIX Ciclo –

SEMI–LAGRANGIAN AND SEMI–IMPLICITDISCONTINUOUS GALERKIN METHODS

FOR ATMOSPHERIC MODELING APPLICATIONS

Presented to Dipartimento di Matematica “F. Brioschi”POLITECNICO DI MILANO

byMarco Restelli

Master Degree in Mechanical Engineering,Politecnico di Milano, Italy

AdvisorProf. Riccardo Sacco

Dipartimento di Matematica “F. Brioschi”,Politecnico di Milano, Italy

CoadvisorDr. Luca Bonaventura

MOX, Modeling and Scientific Computing,Dipartimento di Matematica “F. Brioschi”,

Politecnico di Milano, Italy

Tutor: Prof. Fausto SaleriPhD Coordinator: Prof. Filippo Gazzola

Milano, March 2007

Acknowledgements

First of all I would like to thank my advisor Prof. Riccardo Sacco for his extraordinary support in these yearsand for initiating and guiding me through the field of scientific research, and in particular of applied numericalanalysis.

I am grateful as well to Dr. Luca Bonaventura, co–advisor of this thesis, for introducing me to the subject ofenvironmental modeling, with its many fascinating aspects, for his valuable advices and for his trust.

I am also greatly indebted with Prof. Frank X. Giraldo for his precious support during my extended visitat the Naval Research Laboratory and at the Naval Postgraduate School in Monterey, California, and for theenthusiasm and generosity he shares his scientific interests and ideas with. Prof. Giraldo has been advisingmy work concerning the Semi–Implicit Discontinuous Galerkin formulation described in this thesis and, inparticular, he deserves credit for identifying this research topic.

I would like to thank Dr. Davide Cesari of ARPA–SIM, Emilia Romagna, for his assistance in dealing withvarious atmospheric models.

Fruitful discussions with Drs. V. Aizinger, A. Buffa, C. Carnevale, A. Clappier, C. Dawson, G. Finzi, P.Frolkovic, A. Gassman, M. Giorgetta, C. Lovadina, M. Montgomery, B. Neta, M. Stortini and M. Volta, arealso gratefully acknowledged.

This work would not have been possible without the support of the director of the Modeling and ScientificComputing Group (MOX), Prof. A. Quarteroni, as well as of many members of scientific and technical staffof MOX and of the Mathematical Department ”F. Brioschi”, Politecnico di Milano. I would like to acknowl-edge as well the scientific and technical support I received at the Naval Research Laboratory and at the NavalPostgraduate School in Monterey during my six month visit.

This work has been partially supported by the Italian Ministry of University and Research in the frameworkof the COFIN 2005 program. My visit in Monterey was financially supported by the Office of Naval Re-search Global. The financial support received by CIRM to present preliminary results of this thesis work atthe workshop High Order Non-Oscillatory Methods for Wave Propagation, in Trento, April 2005, is kindlyacknowledged. Preliminary results were also presented in the DWD user seminar in Langen, 2005, in the PDEon the Sphere workshop in Monterey, 2006, as well as in a number of talks given in various places includingUniversita di Pavia and Eidgenossiche technische Hochschule Zurich, whose organizers I would like to thankfor the invitations. Many useful comments were provided by the anonymous reviewers of the paper publishedin Journal of Computational Physics, which essentially corresponds to Chapt. 3 of the present thesis.

Contents

Introduction vii

1. The Semi–Implicit and Semi–Lagrangian Time Discretization Techniques 11.1. The Semi–Lagrangian Approach for the Solution of the Linear Advection Equation . . . . . . 1

1.1.1. Non Conservative Semi–Lagrangian Formulations . . . . . . . . . . . . . . . . . . . 21.1.2. Conservative Semi–Lagrangian Formulations . . . . . . . . . . . . . . . . . . . . . . 4

1.2. The Semi–Implicit Approach for the Time Integration of the Euler Equations . . . . . . . . . 61.2.1. The Basic Semi–Implicit Time Integration Procedure . . . . . . . . . . . . . . . . . . 81.2.2. Semi–Implicit Splitting for the Euler Equations . . . . . . . . . . . . . . . . . . . . . 101.2.3. Semi–Implicit Time Integration of the Euler Equations . . . . . . . . . . . . . . . . . 12

1.3. Semi–Impicit Semi–Lagrangian Time Integration of the Euler Equations . . . . . . . . . . . . 131.4. Coupling SI and SL Time Integration to High Order Finite Element Approaches . . . . . . . . 14

2. The Discontinuous Galerkin Method for PDEs in Conservation Form 152.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2. The Discontinuous Galerkin Method for the Linear Advection Equation . . . . . . . . . . . . 17

2.2.1. Accuracy of the Discontinuous Galerkin Method . . . . . . . . . . . . . . . . . . . . 192.2.2. The Strong Form Discontinuous Galerkin Method . . . . . . . . . . . . . . . . . . . 19

2.3. The Local Discontinuous Galerkin Method for the Heat Equation . . . . . . . . . . . . . . . . 202.4. The Local Discontinuous Galerkin Method for Navier–Stokes Equations . . . . . . . . . . . . 21

2.4.1. Strong form of the Local Discontinuous Galerkin method for the Navier–Stokes equations 23

3. The Semi–Lagrangian Discontinuous Galerkin Method 253.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2. The Semi-Lagrangian Discontinuous Galerkin Method . . . . . . . . . . . . . . . . . . . . . 26

3.2.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2. Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.3. Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.4. The fully discrete SLDG approximation . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3. Linear stability analysis in the one-dimensional case . . . . . . . . . . . . . . . . . . . . . . . 293.4. Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.1. Definition of edge fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.2. Monotonicity of ch,0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.3. Monotonicity of ch,1 and ch,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5. Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.5.1. One–Dimensional Advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.5.2. Two–Dimensional Advection: Solid Body rotation . . . . . . . . . . . . . . . . . . . 363.5.3. Two–Dimensional Advection: Deformational Flow Tests . . . . . . . . . . . . . . . . 413.5.4. Tests on the Advection–Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . 43

3.6. Open Issues and Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4. The Semi–Implicit Discontinuous Galerkin Method 454.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2. The Navier–Stokes Equations for Stratified Flows . . . . . . . . . . . . . . . . . . . . . . . . 464.3. Treatment of the Open Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4. Semi–Implicit Splitting of the Navier–Stokes Equations in Conservation Form . . . . . . . . . 49

4.4.1. Stability Analysis in One Spatial Dimension . . . . . . . . . . . . . . . . . . . . . . 494.5. Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5.1. High–Order Polynomial Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.5.2. Discontinuous Galerkin Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 534.5.3. The Linear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.6. The Fully Discrete Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7. Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.7.1. Collocation Form of the Semi–Implicit Discontinuous–Galerkin Method . . . . . . . . 564.7.2. Efficient Evaluation of The Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.7.3. Filtering the High–Frequency Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.8. Static Condensation of the Momentum Variables by Mass–Lumping . . . . . . . . . . . . . . 60

5. Numerical Validation 655.1. Bubble Convection Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1.1. Density Current Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.1.2. Robert Test Case N 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1.3. Robert Test Case N 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.1.4. Cold Bubble Test Case with Static Condensation . . . . . . . . . . . . . . . . . . . . 69

5.2. Mountain Waves Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2.1. Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2.2. Linear Hydrostatic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.3. Linear Nonhydrostatic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2.4. Linear Irrotational Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.5. Nonlinear Hydrostatic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.6. Nonlinear Nonhydrostatic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6. Conclusions and Future Work 89

Bibliography 91

Introduction

The object of the present PhD thesis is the development and analysis of robust, accurate and efficient numericalmethods for the discretization of partial differential models in environmental fluid dynamics, with particularemphasis on atmospheric flows. Typical examples in this research area can be found in the study of pas-sive/reactive tracer advection in air quality modelling, in local/global Numerical Weather Prediction (NWP)and in climate modelling.

A critical issue in the numerical treatment of such kinds of problems is represented by the stability limitationon the maximum allowable time step that can be used in practical computations, as clearly addressed by thefollowing statement excerpted from [SC91]:

“A long–standing problem in the integration of NWP models is that the maximum permissible time step hasbeen governed by considerations of stability rather than accuracy. For the integration to be stable, the timestep has to be so small that the time truncation error is much smaller than the spatial truncation error, and itis therefore necessary to perform many more time steps than would be the case. The choice of time integrationscheme is, therefore, of crucial importance when designing an efficient weather forecast model, and this is alsotrue when designing environmental emergency response models.”

To cope with the above stringent indications for proper method design, the Semi–Lagrangian (SL) and Semi–Implicit (SI) techniques are highly efficient approaches for the time discretization of equations describingadvection dominated, low Mach number flows. These methods have been developed initially in the contextof numerical weather prediction and their application is credited with a six–fold increase in the computationalefficiency of modern weather prediction systems. The SI and SL time discretization schemes were originallycoupled to finite difference spatial discretizations, and have also been successfully used along with spectraland low order finite element discretizations. Only quite recently an extension of these techniques to modern,high order finite element methods has been proposed. This latter research subject, however, has not yet beenattempted in the case of the Discontinuous Galerkin (DG) finite element method, despite the several advantagesexhibited by such formulation in many applications to high Mach number flows. A first advantage is that theDG method constitutes an extension of finite volume discretization approaches, so that it is naturally tailoredto provide exact discrete local conservation properties. A second advantage is that discontinuous elements arevery flexible to deal with nonconforming grids, and high order accuracy can be achieved even in the presenceof steep gradients without recourse to large computational stencils. This latter feature leads to better scalabilityon massively parallel, distributed memory computer architectures.

Starting from the above general considerations, the present thesis is devoted to investigating the applicability ofSL and SI time discretization techniques in the framework of the DG approaches. This aims to overcome oneof the main shortcomings of the DG method, namely, the severe Courant number limitation associated with theuse of the DG method in combination with explicit Runge–Kutta time stepping, which leads to rather stringentstability restrictions. The methods proposed in this thesis increase substantially the computational efficiencyof DG approaches, especially for applications to atmospheric flow and, more generally, to compressible flowin the low Mach number regime, by consistently incorporating SL and SI time advancing strategies within theDG discretization framework to end up with an accurate, robust and efficient computational procedure.

The main results achieved in this thesis consist in the development, analysis and testing of a novel Semi–

Contents

Lagrangian Discontinuous Galerkin method (SLDG) for the linear advection–diffusion equation and of a Semi–Implicit Discontinuous Galerkin method (SIDG) for the Navier–Stokes Equations for a stratified fluid, respec-tively. In particular, concerning the SLDG method, we notice that:

• the SLDG formulation has been derived for arbitrarily high order elements. The resulting proposedmethod is, to our knowledge, the very first attempt in the literature to combine SL and DG discretizations.The method was proven to be stable independently of the Courant number and was shown to be optimalin many respects for applications to large advection–reaction systems widely present in environmentalmodels;

• a monotonization technique has been derived for a generic DG scheme, based on the Flux CorrectedTransport approach, that greatly reduces numerical diffusion of standard monotonization methods cur-rently in use for DG approximations. This allows to prove a discrete maximum principle for the SLDGmethod, at least in the case of incompressible flows.

Concerning the SIDG method, we notice that:

• for the first time, a high order DG discretization approach has been proposed for the Navier–StokesEquations for a variable density, compressible fluid under the action of gravity;

• a general Semi–Implicit time discretization approach has been derived for this spatial discretization,along the same lines as in traditional SI approaches for finite difference and low order finite elementmethods in geophysical modelling;

• several issues concerning the efficient implementation of the method have been addressed, with particularattention to the derivation of a discrete pseudo–Helmholtz problem for the sole pressure variable by anappropriate static condensation technique, in the same spirit as usually done in finite element hybridizedmethods for the numerical treatment of elliptic problems.

A brief outline of the contents of the thesis reads as follows. In Chapt. 1, the fundamentals of SI and SL dis-cretizations are reviewed, in the context of their application to atmospheric modelling. Chapter 2 provides anoverview of the DG finite element method applied to the discretization of the linear advection equation, the heatequation and the compressible Navier–Stokes Equations. Chapter 3 addresses in detail the description of thenovel SLDG formulation applied to the numerical solution of the linear advection equation. Chapter 4 presentsthe semi–Implicit DG method for the solution of the Navier–Stokes equations in presence of stratification. Thesemi–Implicit DG formulation is then numerically validated and successfully compared in Chapt. 5 with exist-ing established methods on a series of classical test cases. Some concluding remarks and future perspectivesare illustrated in Chapt. 6.

viii

Chapter 1.

The Semi–Implicit and Semi–LagrangianTime Discretization Techniques

In this chapter, a review of the Semi–Implicit and Semi–Lagrangian time discretiza-tion techniques is presented, in the context of their application to atmospheric modeling.Firstly, the linear advection equation is considered. In general, the stability of explicittime integration schemes requires fulfillment of the Courant number condition. The Semi–Lagrangian technique allows for unconditional stability by including the flow character-istics in the numerical scheme. The set of the Euler Equations for compressible stratifiedflows is then considered. For these equations, which constitute the idealized model of anonhydrostatic atmosphere, further limitations on the Courant number are imposed by thefast propagating acoustic and gravity waves. In the context of low Mach number flows,the Semi–Implicit time discretization approach allows for a selective implicit treatmentof the fastest waves, thus increasing the maximum allowable time step while avoidingthe computational cost associated with nonlinear systems typical of fully implicit time–stepping. The combination of the Semi–Implicit and Semi–Lagrangian techniques leadsto efficient and unconditionally stable schemes for the solution of the complete EulerEquations set. Finally, the main goal of the present thesis is stated, namely, the couplingof the Semi–Implicit and Semi–Lagrangian techniques to Discontinuous Galerkin spatialdiscretizations of the linear advection and Euler equations, respectively.

1.1. The Semi–Lagrangian Approach for the Solution of theLinear Advection Equation

The development of accurate and efficient numerical methods for the solution of the Linear Advection Equation(LAE)

∂c

∂t+∇ · (uc) = 0 (1.1)

has always been a main goal of the research on advection dominated flows. In the context of low Mach numberflows, Semi–Lagrangian (SL) formulations are widely acknowledged as an accurate and efficient option. Themain advantage of the SL approach is represented by the combination of the Eulerian and Lagrangian flowdescriptions. As noted in [SC91], in the Eulerian description an observer looks at the world evolve around him

Chapter 1. The Semi–Implicit and Semi–Lagrangian Time Discretization Techniques

at a fixed geographical point, with no explicit reference to the flow streamlines. Numerical methods originatingfrom this viewpoint are subject to the so–called Courant condition [CFL28] (see, for instance, [LeV90]). Inthe Lagrangian description of the flow, on the other hand, an observer looks at the world evolve around himas he travels with a fluid particle, i.e. moving along a streamline. Numerical methods originating from thisviewpoint explicitly refer to the flow streamlines and are not subject to the Courant condition, but require thetracing of fluid particles for long time intervals. Furthermore, fully Lagrangian numerical schemes are not themost straightforward solution for problems on time independent domains. In the SL formulations, a differentset of particles is traced along the flow streamlines at each time step, the sets of particles being chosen so thatthey reach exactly the points of a fixed computational mesh at the end of each time step. By doing so, there isstill as much control on the computational grid as in the Eulerian formulations, while the explicit introductionof the streamlines into the numerical formulation avoids the Courant number condition.

The idea of solving numerically hyperbolic equations using the propagation along characteristics dates backto the work of Courant, Isaacson and Rees [CIR52]. The first general description of the SL method, as op-posed to Eulerian or fully Lagrangian formulations, was presented in [WN59], which generalized some basicideas already contained in [Fjø52]. Although application of the method to operational weather forecastingwas not immediate, an example of three–dimensional SL model is already given in [Kri62]. One of the keysteps in establishing the value of the SL method for meteorological applications is due to [Rob81, Rob82],while at the same time similar methods for the LAE were first proposed also in the mathematical literaturein [Pir82, JR82]. Since then, the approach has been extensively employed, mainly, but not exclusively, formeteorological applications. Full integration of the SL technique with spectral transform models was achievedin [Rit87, RTS+95]. A comprehensive review of the SL formulations can be found in [SC91], while morerecent results are presented, among various references, in [SP92, SG92, FF98]. Various authors have also ex-plored combinations of the SL technique with finite element formulations; in this thesis work, we will referin particular to [MS95] and [Mor98], where the concept of characteristic Galerkin methods is developed. Theissue of cost–effectiveness of the SL approach in alternative to Eulerian formulation is discussed in [McG93]and [BT96]. Presently, the SL method in an essential component of some of the most efficient numericalweather prediction models worldwide, see e.g. [BDP+97, CGM+98, THS01].

1.1.1. Non Conservative Semi–Lagrangian Formulations

In SL methods, equation (1.1) is reformulated in Lagrangian, or advective, form as

dc

dt=∂c

∂t+ u · ∇c = −c ∇ · u (1.2)

and time discretization exploits the fact that the solution values are available along the characteristic lines,which are approximated numerically. In the following, we will consider the LAE (1.2) to be satisfied in anopen bounded domain Ω ⊂ R2 with boundary ∂Ω ≡ Γ and a time interval (0, T ) ⊂ R, with initial and inflowboundary conditions. For the sake of simplicity, it will be assumed that u is a known, smooth and solenoidalvelocity field, i.e., ∇ · u = 0.

In order to describe the time evolution of c(x, t) in (1.2), we define, as in [Mor98], the exact evolution operator

E(t, s) : c(x, t) → [E(t, s)c(·, t)](x) = c(x, t+ s). (1.3)

In order to provide an explicit representation for E we consider that, under mild regularity assumptions on thevelocity field (see [QV94]), it can be proven that streamline or characteristic line functions exist, which aredefined as the solutions of the ordinary differential equations

d

dsX(x, t; t+ s) = u(X(x, t; t+ s), t+ s) (1.4)

2

1.1. The Semi–Lagrangian Approach for the Solution of the Linear Advection Equation

with initial datum at time t given by X(x, t; t) = x. Using the chain rule it is then possible to prove that forany t and s the following relation provides the explicit representation of E

c(x, t+ s) = [E(t, s)c(·, t)](x) = c(X(x, t+ s; t), t). (1.5)

In order to define completely a numerical method for the LAE, a discrete approximation for the evolutionoperator is now required. Let thus J denote a finite set of points xj ∈ Ω, for j = 1, · · · , Npoints, let h ∝ N−1

be a parameter characterizing in some sense the distance between neighboring points, and let us partition thetime interval [0, T ] into a finite number of intervals [tn, tn+1], with tn+1 − tn = ∆t. An approximation ofc(tn, x) in Ω will be denoted by cnh(x). In this section, cnh will be assumed to be known at the grid points xj .

A discrete approximation Ens of E(tn, s), with s ∈ [0,∆t], is completely defined by the following steps:

• a discrete approximation X(x, t; t+ s) for the solution of (1.4)

• an interpolation operator to evaluate c(X(x, t+ s; t), t) from cnh(xj). In the following, the interpolationoperator will be denoted by ΠJ and it will be assumed that ΠJ φh ∈ L∞(Ω) for any grid functionφh(xj), for xj ∈ J .

The unknown at time level tn+1 is then computed as

cn+1h = En

∆tcnh. (1.6)

In the most widely used implementations of the method for application to atmospheric modelling, X is com-puted as

X(x, tn+1; tn) = x− α,

α = u(x− α/2, tn + ∆t/2)∆t(1.7)

and the implicit equation (1.7)2 for α is solved iteratively (see e.g. [SC91]). As discussed in [PS84], substitutingu(x − α/2, tn + ∆t/2) in (1.7) with u(x, tn) to avoid the iterative procedure leads to significant accuracydegradation. On the other hand, explicit trajectory computation can be done by resorting to more sophisticatedapproaches, as discussed in [McG93, Gir99] and [FF98]. In particular, in [FF98] a more abstract framework ispresented, where a generic Runge–Kutta method is used for the solution of (1.4). As shown in [Cas90, RBC05],the time interval ∆t can also be divided into subintervals and then an explicit first order scheme with a smallertime–step can be applied for the solution of (1.4). This choice is more convenient when complicated geometriesare considered, since the substepping effectively prevents the computed trajectories from crossing physicalboundaries.

Concerning the interpolation operator, many alternatives have been explored. When nonuniform Cartesiangrids are considered, an optimal choice is represented by polynomial interpolation with a stencil centered atX . In [SC91] the following interpolation operators are considered: linear, quadratic Lagrange, cubic Lagrange,cubic spline and quintic Lagrange. It is also concluded that cubic interpolation represents a good compromisebetween accuracy and computational cost. This choice gives rise to a fourth–order spatial truncation error withvery little and scale selective damping, mostly affecting the smallest wavelengths. When unstructured grids areadopted, however, the definition of centered interpolation stencils poses significant problems. A typical choiceis thus represented by interpolation with finite element basis functions. Monotonic interpolation operator canalso be considered to improve the stability of the resulting scheme, although this implies the loss of high orderaccuracy [Gir98]

3


The combined effect of trajectory approximation and interpolation on the accuracy of the resulting schemeis analyzed in [FF98]. In this reference, it is shown that, assuming that the solution c(x, t) of (1.2) is suffi-ciently smooth, the Runge–Kutta scheme adopted for the trajectory computation is of order p and that ||c(t)−ΠJ c(t)|J ||L∞(Ω) ≤ e(h) for t ∈ [0, T ], then the following a priori error estimate holds

||c(T )− ch(T )||L∞(Ω) ≤ C

(∆tp +

1∆t

e(h)), (1.8)

where C is a positive constant independent of h and ∆t. Notice that estimate (1.8) suggests the existence of anoptimal and finite value for the time step, and that further reduction of ∆t would result in error increase, whichis confirmed by numerical experiments. We also notice that, for very small time steps, estimate (1.8) can becorrected, following [MS95], as

||c(T )− ch(T )||L∞(Ω) ≤ C

(∆tp +

1∆t

min e(h),M∆t), (1.9)

where M is a positive constant independent of h and ∆t.

The stability of the SL formulation was first examined in the von Neumann sense in [BM82] and [PS84], whereit is shown that the SL method is not subject to the CFL condition. A more general analysis, not restricted toCartesian grids and constant coefficients, can be found in [FF98]. We summarize here the stability analysisfor Eq. (1.2) in the one–dimensional case, constant velocity and for linear interpolation. In this context, theadvection problem reads

ct + u cx = 0 in [0 , L]× [0 , T ]

c(x, 0) = c0(x) x ∈ [0 , L],(1.10)

with periodic initial and boundary conditions. Let Th be a uniform partition of [0 , L] into elements Kj =(xj−1/2 , xj+1/2) of size h, and ∆t denote the time step. Moreover, for each Kj ∈ Th, let xj be the midpointof Kj . The key stability parameter is the Courant number C = u∆t

h , which can be split into its integer andfractional part as C = m+ γ, with m ∈ N and γ ∈ [0 , 1). The solution at time level tn+1 is computed as

cn+1h (xj) = γcnh(xj−m−1) + (1− γ)cnh(xj−m).

Assuming now cnh(xj) = eikj∆x, with i =√−1, k ∈ N, we have

cn+1h (xj) = e−ikm∆t

(γe−ik∆x + (1− γ)

)cnh(xj),

thus yielding the amplification factor e−ikm∆t(γe−ik∆x + (1− γ)

), whose absolute value can be verified to

be less than or equal to 1. This proves the unconditional stability of the scheme.

1.1.2. Conservative Semi–Lagrangian Formulations

In [Ber90] it is shown that, when cubic spline interpolation is adopted on uniform Cartesian grids and the flowis divergence–free, the SL method conserves the tracer mass. However, generic SL formulations as describedin Sect. 1.1.1 are inherently non conservative (see [SC91]). A number of approaches have been proposedto overcome the lack of mass conservation of semi - Lagrangian methods. In many practical applications, aposteriori mass restoration is performed to keep the mass of the atmosphere constant. The methods proposedin [GS94, Pri93] enforce mass conservation as a global constraint, which is achieved via redistribution ofthe mass gains or losses among all mesh points. Neither of these simpler approaches guarantees local massconservation, i.e., changes in the solution at a given mesh point do not necessarily depend only on the values atthe neighbouring mesh points.

4

1.1. The Semi–Lagrangian Approach for the Solution of the Linear Advection Equation

In order to achieve local mass conservation, two main strategies have been pursued. Since both of these strate-gies require the definition of some control volume, we now introduce a partition Th of Ω into Nel triangularelements K.

In the first strategy, the advection equation (1.2) is integrated over a volume Ω(t) that is moving with the flow,in order to obtain

d

dt

∫Ω(t)

c(x, t) dx = 0. (1.11)

Taking Ω(t + ∆t) to coincide with a mesh control volume, as in the non conservative semi - Lagrangianapproach, time integration of (1.11) yields∫

Ω(t+∆t)

c(x, t) dx =∫

Ω(t)

c(x, t) dx, (1.12)

where Ω(t) is now the upstream control volume which evolves into Ω(t + ∆t) within the time step ∆t (seeFig. 1.1). Equation (1.12) is then discretized by approximate reconstruction of the upstream control volumeand approximate computation of the integral on the right hand side. This approach has been sometimes calledconservative remapping, or cell integrated semi–Lagrangian method. The idea of remapping dates back at leastto [HAC79]. Mass conserving variants of the semi - Lagrangian method based on this concept of remappingwere introduced for example in [LP95, MO97, MO98, NM02].

u x0

x∗

K∗

K

Figure 1.1.: The upstream control volume used in advective form conservative extensions of semi–Lagrangianschemes.

ηξ

n

A

Kx

x∗ue

Figure 1.2.: The flux tube used in flux form conservative extensions of semi–Lagrangian schemes.

5


In the second strategy, that is more similar in spirit to Eulerian finite volume methods, equation (1.1) is in-tegrated in space over a fixed mesh control volume Ω and the divergence theorem is applied as usual. Theresulting equation is then integrated in time over a generic time step ∆t, in order to obtain

∫Ω

c(x, t+ ∆t) dx =∫

Ω

c(x, t) dx−∫ t+∆t

t

ds

∫∂Ω

c(ξ, s)u(ξ, s) · n dξ. (1.13)

Equation (1.13) is then discretized by approximate reconstruction of the flux through the domain boundary ∂Ωover the time step ∆t, see Fig. 1.2, where Ω is assumed to coincide with a mesh element K and the flux tubebased on a portion of ∂K is represented. Semi–Lagrangian backward trajectories that reach a number of pointsalong ∂K at time t + ∆t are computed and used for the approximation of the fluid portion that is advectedthrough the boundary. Some form of polynomial reconstruction is then used at these points to discretize thespace time integral on the right–hand side. Methods that can be described in this way will be identified in thefollowing as flux form semi–Lagrangian methods, based on the terminology introduced by S.J. Lin and R. Roodin [LR96], which is probably the most widely known and applied numerical scheme in this group. Examplesof these techniques have been introduced in [DB00, Fey93, Fey98a, Fey98b, Fro02, Fro04, LLM96, LR05].In [Mor98], these approaches were described in a wider context as generalized Godunov methods and a stabilityanalysis was outlined. It should be remarked that flux form semi–Lagrangian methods can also be interpretedas a natural generalization of the so called wave propagation methods, see e.g. [Lev96].

1.2. The Semi–Implicit Approach for the Time Integration of theEuler Equations

The Euler Equations (EE) are generally considered as the basic model for atmospheric motions. Since molec-ular viscosity of air is indeed very small, the viscous effect can be neglected in a first approximation of themost relevant flow regimes, see e.g. [Ped87], while appropriate turbulence closures are necessary for mostpractical applications. The considerations presented in this section are not affected by the introduction of tur-bulent fluxes. Here also the Coriolis force, the diabatic heating of the atmosphere and the presence of moisturewill be neglected for the sake of simplicity. The presentation closely follows [Dur99]; for simplicity, onlytwo–dimensional equations in a vertical plane parallel are considered. The EE in primitive form are given by

∂ρ

∂t= −v · ∇ρ− ρ∇ · v

∂v∂t

= −v · ∇v − 1ρ∇p+ g

∂ei

∂t= −v · ∇ei −

p

ρ∇ · v,

(1.14)

where ρ denotes density, v = [u, w]T is the fluid velocity, p is the pressure, ei = ei(T ) is the internal energy,T is the temperature and g = [0, −g]T is the gravity acceleration. We will also set v = |v|. System (1.14) isclosed by the state equation for ideal gases p = ρRT , where R is the universal gas constant.

In high resolution, nonhydrostatic atmospheric modelling, an equivalent version of the EE is often employed(see e.g. [Cul90, Dur99, Bon00]), in which a change of variables allows for a system of four evolution equationsin four unknowns, thus avoiding the presence of the equation of state. Letting cp, cv denote the specific heat atconstant pressure and volume, respectively, defining as usual γ = cp

cvand denoting by p00 an arbitrarily chosen

reference pressure, we introduce the Exner pressure π =(

pp00

) γ−1γ

and the potential temperature θ = Tπ .

6

1.2. The Semi–Implicit Approach for the Time Integration of the Euler Equations

Sistem (1.14) can now be rewritten as

∂π

∂t= −v · ∇π − R

cvπ∇ · v

∂v∂t

= −v · ∇v − cpθ∇π + g

∂θ

∂t= −v · ∇θ.

(1.15)

In the following, p00 will always denote a reference value for surface pressure, usually taken to be 1000hp. Wenotice that introducing the potential temperature θ as a model variable is particularly attractive in atmosphericapplications, since the quantity dθ

dz directly determines the atmospheric stability (see [Smi79] for details). Fur-thermore, in adiabatic flow potential temperature is proportional to entropy and is exactly conserved along theflow streamlines.

A third form of the EE, the so–called conservation form, can be considered. The main advantage of thisform is that it directly represents the fundamental conservation laws governing the fluid dynamics, namely theconservation of mass, momentum and energy. To obtain this formulation, we need to define the momentumV = ρv, the kinetic energy ec = 1

2ρv2, the potential energy eg = ρgz, z being the vertical coordinate, the

total energy e = ei + ec + eg , the total enthalpy h = e+ pρ , and the energy and enthalpy densities E = ρe and

H = ρh, respectively. With these definitions, system (1.14) can be rewritten as

∂ρ

∂t= −∇ ·V

∂V∂t

= −∇ ·[1ρV ⊗V + p

]+ ρg

∂E

∂t= −∇ ·

[1ρHV

] (1.16)

together with the equation of state.

Finally, for the sake of completeness, we mention here a fourth version of the EE, differing from (1.16) in thatthe energy conservation equation is substituted by an entropy balance equation. Letting Θ = ρθ, system (1.16)can be rewritten as

∂ρ

∂t= −∇ ·V

∂V∂t

= −∇ ·[1ρV ⊗V + p

]+ ρg

∂Θ∂t

= −∇ ·[1ρΘV

].

(1.17)

System (1.17) has been recently adopted for the numerical simulation of atmospheric flows in [KSD00].

After linearizing the problem, it can be verified that the EE admit various types of wave–like solutions, amongwhich acoustic waves can be found. As a consequence, in the explicit numerical integration of this problem,in addition to the Courant number condition associated with the advective terms, analogous conditions have tobe considered for these propagating waves. As a matter of fact, acoustic waves, propagating with velocity a =√γRT ≈ 300 m/s, are generally the fastest signals and result in limitations on the maximum available time–

step which are roughly ten times more restrictive than the limitation associated with advection. To avoid thisrestriction, various approaches have been proposed in the literature. On one hand, modified forms of (1.14) havebeen extensively adopted, which eliminate at the continuous level the acoustic waves via the so called anelasticapproximation (see, for instance, [KE92, Pie84]). This modification of the governing equations, however, cannot be accepted in high resolution mesoscale simulations, since it also affects important flow features [Dal88].Furthermore, stable discretization of anelastic models requires performing a projection step, analogous to what

7


is done in incompressible flow modelling (see e.g. [QV94]). This involves the solution of a Poisson equation,which has approximately the same computational cost as that of a semi–implicit discretization. On the otherhand, so called split explicit discretization techniques have been developed and widely applied in mesoscalemodelling, among which one of the most widespread was introduced in [KW78] and analyzed in [SK92]. Inthis approach, smaller time steps are used to compute the propagation of the fastest propagating waves only,while a longer time step is used for advection and other slower processes, which is only restricted by a Courantcondition based on the advection velocity. Although very effective in practice, split explicit schemes are quitesensitive to the specific way in which the splitting is performed and often require an ad hoc stabilization bymeans of extra numerical diffusion (see e.g. the analysis in [SK92]).

1.2.1. The Basic Semi–Implicit Time Integration Procedure

When the complete set of the EE is adopted, the Semi–Implicit (SI) time integration strategy allows to circum-vent the Courant number condition associated with acoustic waves without introducing a nonlinear implicitproblem. Indeed, in the SI approach, an implicit treatment is selectively applied to the terms responsible for thefastest moving signals, while the remaining terms are treated explicitly. The SI approach was first introducedin [KR70], for the solution of the primitive hydrostatic equations. Modifications were proposed in [Bur75] forthe hydrostatic case and the technique was then extended to the complete non hydrostatic EE in [TW75], withimplicit treatment of the acoustic waves. A similar method has also been applied successfully to the two dimen-sional EE in [CG84]. In [Cul90] and [TRL90], the SI approach as used for the non hydrostatic EE equationsand both acoustic and gravity waves were treated implicitly. Finally, the SI approach has been successfullyapplied to oceanic and coastal modelling, see e.g. [Cas90, CC94, RSL98, MQS99, CZ02].

To introduce the SI time integration procedure, we first consider a linearization of (1.15). We thus define aconstant reference state π, u, w, θ, such that

u = const; w = 0;

∂π

∂x= 0;

∂θ

∂x= 0;

cpθdπ

dz= −g.

(1.18)

Notice that, in particular, (1.18) requires that the reference state is in hydrostatic balance, thus being an equi-librium state. We now let π = π + π′, u = u + u′, w = w′ and θ = θ + θ′ and consider the followingproblem:

∂π′

∂t= −u∂π

′

∂x− dπ

dzw′ − R

cvπ

(∂u′

∂x+∂w′

∂z

)∂u′

∂t= −u∂u

′

∂x− cpθ

∂π′

∂x

∂w′

∂t= −u∂w

′

∂x− cpθ

∂π′

∂z+ g

θ′

θ

∂θ′

∂t= −u∂θ

′

∂x− dθ

dzw′.

(1.19)

Following for instance [Bon00], we discretize in space problem (1.19) by considering finite differences on astaggered C-grid. The time integration is then performed with an operator splitting approach: first the advec-tion is computed with the explicit in time–upwind in space method, yielding (π∗, u∗, w∗, θ∗)(x, z), then thecentered in space–implicit tin time Crank–Nicholson scheme is applied to the remaining terms. The stabilityanalysis of the resulting scheme can be performed following [Str89], Chapts. 2 and 7. We first consider thefrozen coefficient problem obtained by fixing all the coefficients depending on the reference state at a genericpoint of the computational domain, then we substitute (π′, u′, w′, θ′)(x, z, tn) = (πn, un, wn, θn)ei(kx+lz)

8


and (π∗, u∗, w∗, θ∗)(x, z) = (π∗, u∗, w∗, θ∗)ei(kx+lz) for (k, l) ∈ [−π/∆x, π/∆x] × [−π/∆z, π/∆z]obtaining

(π∗, u∗, w∗, θ∗) = (1−Ψ) (πn, un, wn, θn) (1.20)

where Ψ = C(1− e−ik∆x

), with C = u∆t

∆x denoting the advective Courant number, and

πn+1 = π∗ −∆tdπ

dzχl

(αwn+1 + (1− α)w∗

)−∆t

Rπ

cv

(iφk

(αun+1 + (1− α)u∗

)+ iφl

(αwn+1 + (1− α)w∗

))un+1 = u∗ −∆tcpθiφk

(απn+1 + (1− α)π∗

)wn+1 = w∗ −∆tcpθiφl

(απn+1 + (1− α)π∗

)+ ∆t g

θ

(αθn+1 + (1− α)θ∗

)θn+1 = θ∗ − dθ

dz

(αwn+1 + (1− α)w∗

)(1.21)

where α ∈ [0, 1], χl = cos l∆z2 , φk = sin k ∆x

2∆x2

, and φl = sin l ∆z2

∆z2

. The amplification matrix of the scheme is

(A− α∆tP )−1(B + (1− α)∆tP )

withA = I + α∆tM, B = (1−Ψ) (I − (1− α)∆tM) , (1.22)

and where M and P are defined as

M =

0 Rπ

cviφk

Rπcv

iφl 0cpθiφk 0 0 0cpθiφl 0 0 − g

θ

0 0 dθdz 0

, P =

0 0 −dπ

dz χl 00 0 0 00 0 0 00 0 0 0

.Introducing now the diagonal matrix

E =

√cpθ 0 0 0

0√

Rπcv

0 0

0 0√

Rπcv

0

0 0 0 i

√Rπcv

g

θdθdz

it can be verified that EME−1 is an imaginary symmetric matrix, which is then diagonalizable and has purelyimaginary eigenvalues. Therefore, M can be diagonalized by some invertible matrix T and will also havepurely imaginary eigenvalues. Furthermore, it is easy to see that also A and B will be diagonalized by T .Denoting now by iµ the generic eigenvalue of M , where µ is a real number, it follows that the eigenvalues λ ofA satisfy

λ = 1 + iαµ∆t,

so that |λ| ≥ 1, A is invertible and ||A−1|| ≤ 1 uniformly in k∆x, l∆z, ∆x, ∆z and ∆t, || · || denotingthe matrix norm induced by the Euclidean vector norm. It follows that there exists ∆t > 0 such that, for∆t ∈ [0, ∆t], the matrix A− α∆tC is invertible and the following identity holds(

I − α∆tA−1P)−1

= I + α∆tA−1P(I − α∆tA−1P

)−1. (1.23)

The amplification matrix can thus be written as

A−1B + ∆t(αA−1P

(I − α∆tA−1P

)−1A−1B + (1− α)A−1P

+∆tα(α− 1)A−1P(I − α∆tA−1P

)−1A−1P

).

9


To control the terms into parenthesis, we use the following estimates, uniform in k∆x, l∆z, ∆x, ∆z and ∆t:

||A−1P || ≤ ||A−1|| ||P || ≤∣∣∣∣dπdz

∣∣∣∣||(I − α∆tA−1P

)−1 || ≤ 11− α∆t||A−1|| ||P ||

≤ 11− α∆t

∣∣dπdz

∣∣ . (1.24)

From standard results in the stability theory of finite difference schemes (see [Str89]), the necessary and suffi-cient condition for stability is given by ||A−1B|| ≤ 1 +O(∆t). This is equivalent to proving that the solutionsof the generalized eigenvalue problem det(B − λA) = 0 belong to the unit circle up to O(∆t), and since bothB and A are diagonalized by the same matrix T we have

λ = (1−Ψ)1− i(1− α)µ∆t

1 + iαµ∆t.

We first consider the case u = 0. In this case, it can be verified that

|λ|2 = 1− (2α− 1)µ2∆t2

1 + α2µ2∆t2

so that the scheme is unconditionally unstable for α < 1/2 and unconditionally stable for α ≥ 1/2. If we nowlet u > 0, we obtain

|λ|2 = |1−Ψ|2(

1− (2α− 1)µ2∆t2

1 + α2µ2∆t2

)so that a sufficient condition for stability is |1 − Ψ| ≤ 1, which is equivalent to 0 ≤ C ≤ 1. A similaranalysis can be performed when a multistep time–integration scheme is considered, as discussed, for instance,in [Dur99].

For the sake of completeness, we now compute the eigenvalues µ. These are the solutions of the equation

µ4 −(a2(φ2

k + φ2l

)+N2

)µ2 + a2N2φ2

k = 0 (1.25)

where we have introduced the Brunt–Vaisala frequency N2 = g

θdθdz and the sound speed a2 = γRT . We get

µ2 =a2

2

φ2k + φ2

l +N2

a2±

[(φ2

k + φ2l +

N2

a2

)2

− 4N2φ2k

a2

]1/2 . (1.26)

For usual meteorological applications, it can be assumed that2aNφk

a2(φ2k + φ2

l ) +N2 1, yielding

µ21 = a2

(φ2

k + φ2l +

N2

a2

), µ2

2 =N2φ2

k

φ2k + φ2

l +N2

a2

(1.27)

with µ22 µ2

1.

1.2.2. Semi–Implicit Splitting for the Euler Equations

In this section we describe a generalization of the SI time integration strategy discussed in Sect. 1.2.1 to non-linear problems. This will provide the general framework for the application of the method to the completeEE. We notice here that, although the SI approach is usually considered in combination with either the Crank–Nicolson or the leapfrog time integration schemes, following the abstract formulation of [Gir05] it is possibleto include other time integration schemes, such as the backward difference scheme discussed in the sequel.

10


Given the ODEdqdt

= S (q), (1.28)

we define an affine operator A G such that A Gq ≈ S (q) and set

dqdt

=S (q)−A Gq

+ A Gq. (1.29)

The main idea is now to treat the term in braces explicitly, while the remaining terms will be treated implicitly.In mesoscale atmospheric modeling, moreover, it is common to introduce some artificial dissipative terms tohandle the open boundary conditions and to increase the stability of the numerical formulation (this will beextensively discussed in Capt. 4). For these terms, accuracy is not an issue, while it is important to ensurestrong numerical damping. To account for these terms, it is convenient to split the affine operator A G asA G = A + A 0, and set A q = L q + f and A 0q = L 0q + f0. Eq. (1.29) can now be written as

dqdt

=

S (q)−L q

+ L q + L 0q + f0, (1.30)

where S (q) = S (q)−A 0q.

The discretization of (1.30) can now be constructed as follows:

dqdt

≈ 1γ∆t

[qn+1 −

2∑m=0

αmqn−m

]

S (q)−L q

≈2∑

m=0

βm

(S (qn−m)−L qn−m

)L q ≈

2∑m=−1

σmL qn−m

L 0q ≈ L 0qn+1

(1.31)

for suitable coefficients αm, βm and σm. Notice that, for consistency, it is required that∑m

αm =∑m

βm =∑m

σm = 1. (1.32)

Typically, the operator L is chosen in such a way that, for a particular range of q, the term S −L vanishes,and time integration is performed with the implicit scheme (1.31)1,3,4. In Tab. 1.1 it is shown how to recoversome classical time marching schemes by properly choosing the coefficients in (1.31).

After some algebraic manipulations, exploiting the fact that L is linear and does not depend on the time leveltn, we obtain the following implicit scheme

qn+1 = qex + γ∆t

[2∑

m=−1

ρmL qn−m + L 0qn+1 + f0

], (1.33)

where ρm = σm − βm, β−1 = 0 and

qex =2∑

m=0

αmqn−m + γ∆t2∑

m=0

βmS (qn−m). (1.34)

Notice that from (1.32) it follows that∑

m ρm = 0. Finally, letting

qtt =2∑

m=−1

ρmqn−m

q∗ = ρ−1qex +2∑

m=0

ρmqn−m,

11


Method α0 α1 α2 γ β0 β1 β2 σ−1 σ0 σ1 σ2

CN 1 0 0 1 1 0 0 θ 1− θ 0 0LF2 0 1 0 2 1 0 0 θ 0 1− θ 0BDF2a 4/3 −1/3 0 2/3 2 −1 0 1 0 0 0BDF2b 4/3 −1/3 0 2/3 8/3 −7/3 2/3 1 0 0 0

Table 1.1.: Crank–Nicolson (CN), leapfrog (LF2), and backward difference (BDF2a and BDF2b) time integra-tion schemes and their associated coefficients in the context of Eq. (1.31).

we obtain

(I − γ∆tL 0

)qtt = q∗ + ρ−1γ∆tL qtt − γ∆tL 0

(2∑

m=0

ρmqn−m

)+ ρ−1γ∆tf0. (1.35)

It is now convenient, for implementation purposes, to introduce B =(I − γ∆tL 0

)−1, thus obtaining

qtt = Bq∗ + ρ−1 (γ∆tBL )qtt −(γ∆tBL 0

)( 2∑m=0

ρmqn−m

)+ ρ−1

(γ∆tBf0

), (1.36)

where it should be noted that the computation of the following operators and constant term is required: B,(γ∆tBL ),

(γ∆tBL 0

)and

(γ∆tBf0

).

Summarizing, the semi–implicit method requires the following steps (see equations (1.34) and (1.36)):

• qex =2∑

m=0

αmqn−m + γ∆t2∑

m=0

βmS (qn−m);

• q∗ = ρ−1qex +2∑

m=0

ρmqn−m;

• (I − ρ−1 (γ∆tBL ))qtt = Bq∗ −(γ∆tBL 0

)( 2∑m=0

ρmqn−m

)+ ρ−1

(γ∆tBf0

);

• qn+1 =1ρ−1

(qtt −

2∑m=0

ρmqn−m

).

Application of the semi–implicit splitting to PDEs requires, in addition to the previous steps, the definition ofdiscrete counterparts of the operators S , L and L 0, which will be denoted by Sh, Lh and L 0

h , respectively.If Lh and L 0

h are also linear, relations (1.30) and (1.31) hold for the discretized operators as well. In this case,the fully discretized scheme could be obtained performing the spatial semidiscretization first and then applyingthe semi–implicit time discretization splitting to the resulting ODE.

1.2.3. Semi–Implicit Time Integration of the Euler Equations

In this section we exploit the general formulation introduced in Sect. 1.2.2 to discretize in time problem (1.15).To illustrate the role of the operator L 0 introduced in (1.30), we also include in the presentations the terms

12

1.3. Semi–Impicit Semi–Lagrangian Time Integration of the Euler Equations

associated with open boundary conditions. Let thus πb, ub, wb and θb denote prescribed boundary data onthe open boundaries of the computational domain and, following a well established approach in atmosphericmodeling (see for instance [KW78]) consider the following problem:

∂π

∂t= −u∂π

∂x− w

∂π

∂z− R

cvπ

(∂u

∂x+∂w

∂z

)− τπ(π − πb)

∂u

∂t= −u∂u

∂x− w

∂u

∂z− cpθ

∂π

∂x+ µx − τ(u− ub)

∂w

∂t= −u∂w

∂x− w

∂w

∂z− cpθ

∂π

∂z− g + µz − τ(w − wb)

∂θ

∂t= −u∂θ

∂x− w

∂θ

∂z− τ(θ − θb).

(1.37)

In (1.37), τπ and τ are nonnegative coefficients which are zero far from the domain boundary and positivewithin a fixed distance from the boundary itself. The region where τ and τπ are positive is referred to asabsorbing or sponge layer.

To apply the SI time integration scheme to problem (1.37), it will suffice to define the linear operators L andL 0 in such a way that, for small deviations from a suitable reference state, the stable time discretization (1.21)is recovered. This can be accomplished by first identifying a reference state as in (1.18) and then setting

L q =

−dπdzw − R

cvπ

(∂u

∂x+∂w

∂z

)−cpθ

∂π

∂x

gθ

θ− cpθ

∂π

∂z

−dθdzw

f =

0

0

−2g

0

(1.38)

and

L 0q = −

τπ π

τ u

τ w

τ θ

f0 =

τπ πb

τ ub

τ wb

τ θb

. (1.39)

Notice that the SI time discretization, and in particular Eq. (1.36), is well defined even in the limit of τπ, τ →∞. Indeed, it can be verified that formally setting τπ = τ = +∞ in (1.39) amounts to incorporating theboundary conditions into the time marching scheme.

1.3. Semi–Impicit Semi–Lagrangian Time Integration of theEuler Equations

From the considerations discussed in the previous sections, it is easy to see that the SL and SI time dis-cretization approaches can increase greatly the computational efficiency of numerical models for applicationsto atmospheric modelling. Indeed, combining the SI and SL approaches allows to obtain an uncondition-ally stable scheme for the EE, thus enabling to choose the time step based on accuracy considerations only,rather than on the basis of a stability restriction. This fact was first emphasized in the seminal papers by A.Robert [Rob81, Rob82], in which the great potential of this combination was demonstrated for the first time.Presently, various operational models for atmospheric applications rely on a SI–SL formulation, among which

13


we mention [TRL90, BDP+97] and [CGM+98], where finite differences are adopted for the space discretiza-tion, and [THS01], where the space discretization employs the spectral method.

To introduce the SL treatment of the advective terms we proceed as in Sect. 1.2.1 substituting a semi–Lagrangiansubstep to the explicit in time–upwind in space first step. This amounts to setting π∗h = En

∆tπ′nh , and analogous

expression for u∗h, w∗h and θ∗h, where En∆t was defined in Sect. 1.1.1. As a consequence, the term 1−Ψ in (1.22)

is substituted by e−imk∆x(1− γ

(1− e−ik∆x

)), where m and γ denote the integer and fractional part of the

advective Courant number C. The resulting scheme is thus unconditionally stable.

1.4. Coupling Semi–Implicit and Semi–Lagrangian TimeIntegration to High Order Finite Element Approaches forSpatial Discretization

The SI and SL time discretization techniques have been developed originally coupled to finite difference spatialdiscretizations. Subsequently, also spectral and low order finite element discretizations were coupled success-fully to these approaches. Only quite recently the extension of these techniques to advanced, high order finiteelement methods has been developed, see e.g. [Gir98, Gir00b, GHW02, Gir05]. However, so far this has notbeen attempted for DG approaches.

On the other hand, numerous potential advantages of the DG method have been highlighted in the applica-tions to high Mach number flows. The DG method constitutes an extension of finite volume discretizationapproaches, so that it allows naturally for exact discrete conservation properties. Discontinuous elements arevery flexible for applications on nonconforming grids. High order accuracy can be maintained even whensteep gradients are present without recourse to large computational stencils. This leads to better scalability onmassively parallel, distributed memory computer architectures.

For this reason, the present thesis is devoted to investigating the applicability of SL and SI time discretizationtechniques in the framework of the DG approaches. In fact, the DG method is usually employed in combinationwith explicit Runge-Kutta time stepping, which leads to rather stringent stability restrictions. Thus, one of themain aims of the present work is to increase substantially the computational efficiency of DG approaches,especially for applications to atmospheric flow and, more generally, to compressible flow in the low Machnumber regime, by application of these efficient and robust time discretization techniques.

14

Chapter 2.

The Discontinuous Galerkin Method forPartial Differential Equations inConservation Form

In this chapter we review the basic formulation of the Discontinuous Galerkin methodfor the solution of partial differential equations in conservation form. In Sect. 2.1 the stan-dard notation for the presentation of Discontinuous Galerkin formulations is introduced.As the interest of our work is in both hyperbolic and elliptic problems, this section borrowsthe notation usually adopted in these two different contexts. In Sect. 2.2, the applicationof the Discontinuous Galerkin method to the linear advection equation is presented, whilein Sect. 2.3 the heat equation is considered. Finally, in Sect. 2.4 the formalism introducedis combined to handle both the hyperbolic and elliptic components of the Navier–Stokesequations.

The Discontinuous Galerkin (DG) method originates from a Galerkin approximation of the continuous prob-lem where the conservative form of the equations is considered and discontinuous basis functions are adopted.A posteriori, it can be verified that the DG method actually represents a generalization to arbitrary order ofthe finite volume method. Stemming from the conservative form of the continuous problem, the DG methodprovides discrete balance equations that reproduce at the element level the fundamental physical balance lawscharacterizing the continuous problem. This makes the DG method a natural choice for problems whose solu-tion presents discontinuities (where the non conservative formulation does no longer hold, see [Tor97, LeF02]),and experimental results show that even when the solution is smooth but with steep gradients the DG methodis superior to the continuous Galerkin method, for a given spatial resoltion. Further advantages are the highflexibility, allowing for nonconforming hp adaptivity, and good scalability on parallel architectures, thanks tothe limited computational stencil. On the other hand, since the continuity constraint is eliminated, for a givenspatial resolution the DG formulation requires a larger number of degrees of freedom than a continuous method,thus resulting, in general, in a computationally more expensive technique. We notice, however, that recentlysome attempts have been made to revisit the DG method in order to obtain a formulation whose computationalcost in not larger than the analogous continuous formulation [HSBB06].

The DG method for the solution of steady state hyperbolic problems was first introduced in [RH73] for thesolution of the neutron transport equation. The first analysis of the DG method is presented in [LR74], whereit is proven that the method has a convergence rate of O(hk) for general triangulations and O(hk+1) forCartesian grids, k ≥ 0 denoting the degree of the polynomial finite element space used in the approximation.The first estimate was then refined in [JP86] where the convergence rate O(hk+1/2) was proven for a general

Chapter 2. The Discontinuous Galerkin Method for PDEs in Conservation Form

triangulation, and this estimate was shown to be optimal in [Pet91].

The application of the DG method to time dependent problem was first attempted in [CC89], where explicitforward Euler time stepping is stabilized with a local projection operator based on the monotonicity preservingprojection introduced by Van Leer in [Lee74]. The approach is refined in [CS91], where second and third orderRunge Kutta explicit time integration schemes are introduced, thus yielding the Runge Kutta–DiscontinuousGalerkin (RKDG) method. In this formulation, the projection operator is no longer required to ensure stabilityin the linear case, however it is retained to ensure monotonicity of the computed solution. It is also proven that,for constant Courant number, the RKDG method is formally uniformly (k + 1)st–order accurate in time andspace. The RKDG method is then extended to systems of equations and multidimensional problems in [CS89,CL89, CHS90, CS98].

The application of discontinuous finite element formulations to elliptic and parabolic problems has been stud-ied somewhat independently from the development of DG formulations for hyperbolic conservation lawsin [Nit71, Whe78, Arn82]. Various formulations have been proposed since then, the main difference beingthe way the diffusive flux is described at the interelement boundaries. In particular, we mention here theInterior Penalty method introduced in [Whe78], the Baumann–Oden method described in [BO99], the NonSymmetric Interior Penalty (NIPG) presented in [RWG99], the Bassi–Rebay method of [BR97a] and the LocalDiscontinuous Galerkin method (LDG) proposed in [CS98]. The various formulations were then analized ina unified framework in [ABCM02], while a convergence analysis in the general case of hp nonconformingadaptivity was presented in [HSS02].

Finally, the DG method has been applied to a wide class of viscous and inviscid flows. Here, we mention thework of Bassi and Rebay for the solution of the Euler equations and the Navier–Stokes equations [BR97b,BR97a] and the applications of the DG method for the simulation of environmental flows by Dawson andAizinger [DA05] and by Giraldo [GHW02, GW].

2.1. Notation

Let Ω be an open bounded domain in R2 and Th denote a partition of Ω into non overlapping elements K, with

h = maxK∈Th

diam(K).

For the sake of simplicity, we will assume that periodic boundary conditions are prescribed. The boundary ofan element is denoted by ∂K and the outward unit normal vector by n∂K , moreover the notation ne,∂K willbe employed to indicate the restriction of n∂K on the edge e ⊂ ∂K. The characteristic function of an elementK will be denoted by 1K . Let as well [0, T ] denote a bounded time interval partitioned into N subintervals[tn, tn+1] with t0 = 0, tn+1 = tn + ∆t and tN = T .

Let k be a nonnegative integer; then Vk(K) = Pk(K) denotes the space of polynomials of degree at most k onelement K, with Nk ≡ dim(Vh(K)) = 1

2 (k + 1)(k + 2). The space of elements of L∞(Ω) whose restrictionto K ∈ Th belongs to Vk(K) is denoted by Vh. Notice that functions in Vh are in general discontinuous acrosseach edge e. For a given element K, edge e ⊂ ∂K, point ξ ∈ e and vh ∈ Vh, we can thus define

vh(ξint(K)) = limx→ ξ

x ∈ K

vh(x), vh(ξext(K)) = limx→ ξ

x /∈ K

vh(x).

This definition can be extended to vector valued functions in (Vh)n by applying it componentwise. For e =∂K ∩ ∂K ′, ξ ∈ e, the average and jump for a scalar function vh ∈ Vh and vector function rh ∈ (Vh)2 are

16

2.2. The Discontinuous Galerkin Method for the Linear Advection Equation

defined respectively as follows:

vh (ξ) =12

(vh(ξint(K)) + vh(ξint(K′))

)rh (ξ) =

12

(rh(ξint(K)) + rh(ξint(K′))

)[[vh]](ξ) = vh(ξint(K))ne,∂K + vh(ξint(K′))ne,∂K′ [[rh]](ξ) = rh(ξint(K)) · ne,∂K + rh(ξint(K′)) · ne,∂K′ .

The L2 projector onto Vh will be denoted by Πh.

2.2. The Discontinuous Galerkin Method for the LinearAdvection Equation

The following presentation closely follows [CJST97]. In this section, we address the DG discretization of theLAE, repeated here for the sake of completeness

∂c

∂t+∇ · (uc) = 0 inΩ× (0, T ) (2.1)

with suitable initial and boundary conditions. Here, u is assumed to be a known smooth function u : Ω ×[0, T ] → R2. Notice that the conservative form of the LAE is considered, this being essential in order to obtaina discrete conservation property. An approximation ch = ch(x, t) to the solution c(x, t) of (2.1) is sought,such that ch ∈ Vh at each time level. Multiplying (2.1) by a function vh ∈ Vh, integrating over K ∈ Th andreplacing the exact solution by its approximation ch, we get

d

dt

∫K

ch(x, t)vh(x)dx = −∫

K

∇ · (u(x, t)ch(x, t)) vh(x)dx ∀vh ∈ Vh. (2.2)

Then, formally integrating by parts, we obtain

d

dt

∫K

ch(x, t)vh(x)dx =∫

K

ch(x, t)u(x, t) · ∇vh(x)dx−∫

∂K

ch(ξ, t)u(ξ, t) · n∂Kvh(ξ)dξ ∀vh ∈ Vh.

(2.3)Notice that the advective boundary term ch(ξ, t)u · n∂K does not yet have a precise meaning, because chis a discontinuous function across interelement boundaries. In order to resolve this ambiguity, the upwindnumerical flux he,K(ch) = he,K(ch(ξint(K), t), ch(ξext(K), t)) is introduced to replace ch(ξ, t)u · n∂K , ξ ∈e ⊂ ∂K, in the boundary integrals. The DG formulation of the LAE thus reads:find ch(x, t) such that ch(·, t) ∈ Vh for all t ∈ [0, T ] and

d

dt

∫K

ch(x, t)vh(x)dx =∫

K

ch(x, t)u(x, t) · ∇vh(x)dx−∑

e∈∂K

∫e

he,K(ch(ξ, t))vh(ξ)dξ

∀vh ∈ Vh, ∀K ∈ Th.

(2.4)

The upwind flux he,K is defined as

he,K(a, b) = u · ne,∂Ka+ b

2− |u · ne,∂K |

b− a

2. (2.5)

Notice that, for e = ∂K ∩ ∂K ′, the following property holds:

he,K(ch(ξint(K), t), ch(ξext(K), t)) = −he,K′(ch(ξint(K′), t), ch(ξext(K′), t)). (2.6)

As a consequence, the introduction of the numerical flux into the discrete weak one–element formulationamounts to enforcing in weak form a boundary condition in the one–element problem (2.3).

17


Remark 2.2.1 We notice that the upwind flux (2.5) can also be expressed in terms of the streamlines andevolution operator introduced in Sect. 1.1.1 as

he,K(ch) = u · ne,∂K lims→0−

ch(X(ξ, t; t+ s), t+ s) (2.7)

= u · ne,∂K lim∆t→0+

[E(t−∆t,∆t)ch(·, t−∆t)] (ξ).

This expression will be useful in view of combining the semi–Lagrangian time discretization with the DGformulation in Chapt. 3.

Equations (2.4) and (2.5) represent the DG spatial discretization of the LAE. In compact notation, this can berepresented as

d

dtch = Lh(ch), (2.8)

and a comparison with (2.1) shows that Lh(ch) ≈ −∇ · (uch).

To compute the integrals in (2.4) numerical quadrature rules are used. In [CHS90] it is suggested that thequadrature rule for the edge integrals should be exact for polynomials of degree 2k+1 and that for the elementintegrals should be exact for polynomials of degree 2k. An alternative choice for the case of a high order DGformulation will be discussed in Chapt. 4.

The discrete conservation properties of the DG method can be immediately verified from (2.4). As a matter offact, thanks to the discontinuous nature of vh, it is possible to set vh = 1K , thus obtaining

d

dt

∫K

ch(x, t)dx = −∑

e∈∂K

∫e

he,K(ch(ξ, t))dξ,

which, in virtue of (2.6), expresses a local discrete conservation law.

Problem (2.4) constitutes an ordinary differential equation still requiring a time discretization. This is usu-ally done by means of the Total Variation Diminishing (TVD) Runge–Kutta schemes proposed in [Shu88].Moreover, in order to ensure a TVD property of the computed solution, which in turn ensures convergenceto a weak solution by the Lax–Wendroff Theorem [LW60], a projector operator is introduced. This projec-tor, denoted by ΛΠh, is a generalization of the slope limiter operator introduced in [Lee74] and is such that∫

KΛΠhch =

∫Kch, which is a crucial element to preserve the discrete conservation property of the DG formu-

lation. Denoting by c0 the initial condition for the LAE, the complete RKDG method can thus be summarizedby the following steps:

• set c0h = Πhc0;

• For n = 0, · · · , N − 1 compute cn+1h from cnh as follows:

1. set c(0)h = cnh;

2. for i = 1, . . . , k + 1 compute the intermediate functions:

c(i)h = ΛΠh

i−1∑l=0

αilc(l)h + βil∆tnLh(c(l)h )

;

3. set cn+1h = c

(k+1)h .

18

2.2. The Discontinuous Galerkin Method for the Linear Advection Equation

In [CS91] it is shown that, for k = 1 and one spatial dimension, the RKDG scheme is L∞(0, T ;L2(0, 1))stable provided the following CFL condition holds

u∆th≤ 1

3.

For higher polynomial orders, numerical experiments indicate the following CFL condition

u∆th≤ 1

2k + 1. (2.9)

2.2.1. Accuracy of the Discontinuous Galerkin Method

Concerning the accuracy of the RKDG method, we mention here the following results:

• In [JP86] a steady state linear problem is considered and the following error estimate is obtained:

||c− ch||L2(Ω) ≤ Chk+ 12 , (2.10)

where C is a positive constant. Notice that the rate of convergence guaranteed by this result is less thanthe optimal rate O(hk+1).

• The optimal convergence rate O(hk+1) has been obtained in [LR74] for meshes consisting entirely ofrectangles and in [Ric88] under the assumption that all the element edges are bounded away from thecharacteristic directions.

• It has been an open question whether the previous additional assumptions on the computational grid arenecessary to obtain an estimate of optimal order. In [Pet91] a counterexample is presented, indicatingthat (2.10) is sharp. On the other hand, the alignment of the element edges with the characteristic direc-tions seems to be a necessary condition to observe the convergence rate degradation. Since this is a ratherunusual situation, it can be expected that in practical numerical experiments the optimal order O(hk+1)might be observed.

• In [CHS90] time dependent problems are considered, and the notion of formal accuracy is introduced as

|| − ∇ · c− Lh(c)||L∞(Ω) ≤ Chk+1|c|W k+2,∞(Ω). (2.11)

Numerical experiments are also presented showing that both ||c(T )−ch(T )||L1(Ω) and ||c(T )−ch(T )||L∞(Ω)

are proportional to hk+1.

• In [CS98] the following error estimate is proven for the time dependent linear problem:

||c(T )− ch(T )||L2(Ω) ≤ Chk+ 12 , (2.12)

which is in agreement with (2.10).

2.2.2. The Strong Form Discontinuous Galerkin Method

Besides (2.4), an alternative formulation, which is referred to as strong form DG in [GHW02] and integral formin [CHQZ06], is sometimes employed (see also [LK99]). This is obtained by further integrating by parts the

19


second term in (2.4), yielding the problem:find ch(x, t) such that ch(·, t) ∈ Vh for all t ∈ [0, T ] and

d

dt

∫K

ch(x, t)vh(x)dx = −∫

K

∇ · (u(x, t)ch(x, t)) vh(x)dx (2.13)

+∑

e∈∂K

∫e

(ch(ξ, t)u(ξ, t) · n∂K − he,K(ch(ξ, t))) vh(ξ)dξ

∀vh ∈ Vh, ∀K ∈ Th.

In (2.13) it can be seen that the element boundary term actually introduces a penalization of the jumps of ch.

The two DG formulations (2.4) and (2.13) are equivalent as long as all the integrals are computed exactly, whilethey lead to (slightly) different results when approximate quadratures are introduced. Numerical experimentsin [GHW02] indicate that the strong form can result in a greater accuracy. This is the reason why it will beadopted in Chapt. 4.

A remark is however necessary concerning the conservation properties of the strong form DG method. As amatter of fact, setting vh = 1K in (2.13) yields

d

dt

∫K

ch(x, t)dx = −∫

K

∇ · [u(x, t)ch(x, t)] dx (2.14)

+∑

e∈∂K

∫e

(ch(ξ, t)u(ξ, t) · n∂K − he,K(ch(ξ, t))) dξ,

which can not not immediately be interpreted as a conservation statement. If now the quadrature formula ischosen in such a way that these integrals are computed exactly, counterintegration by parts of (2.14) gives (2.2),thus ensuring discrete conservation. We notice that, in particular, if u is uniform, choosing the quadratureformula as suggested in [CHS90] guarantees discrete conservation.

2.3. The Local Discontinuous Galerkin Method for the HeatEquation

In this section, we describe the Local Discontinuous Galerkin (LDG) formulation for the heat equation. Weclosely follow [CS98] and [CCPS00], considering first the linear case, and postponing the generalization tononlinear problems to Sect. 2.4.

To introduce the LDG method, we first rewrite the heat equation introducing the auxiliary variable σ∂c

∂t+∇ · (−aσ) = 0 inΩ× (0, T )

σ −∇c = 0 in Ω× (0, T ),(2.15)

with suitable initial and boundary conditions. Here, a is a known smooth scalar function. An approximation(ch,σh) = (ch(x, t),σh(x, t)) to the solution (c(x, t),σ(x, t)) of (2.15) is sought, such that (ch,σh) ∈ Vh ×(Vh)2 at each time level. Multiplying the two equations (2.15) by test functions vh ∈ Vh and rh ∈ (Vh)2,respectively, integrating over K ∈ Th and replacing the exact solution by its approximation, we get

d

dt

∫K

ch(x, t)vh(x)dx+∫

K

∇ · (−aσh(x, t)) vh(x)dx = 0 ∀vh ∈ Vh∫K

σh(x, t) · rh(x)dx−∫

K

∇ch(x, t) · rh(x)dx = 0 ∀rh ∈ (Vh)2.(2.16)

20

2.4. The Local Discontinuous Galerkin Method for Navier–Stokes Equations

Then, formally integrating by parts and introducing the numerical fluxes (c, σ), in analogy with the hyperboliccase, we obtain the following problem:find (ch(x, t),σh(x, t)) such that (ch(·, t),σh(·, t)) ∈ Vh × (Vh)2 for all t ∈ [0, T ] and

d

dt

∫K

ch(x, t)vh(x)dx−∫

K

(−aσh(x, t)) · ∇vh(x)dx+∫

∂K

(−aσ(ξ, t)) · n∂Kvh(ξ)dξ = 0 ∀vh ∈ Vh∫K

σh(x, t) · rh(x)dx+∫

K

ch(x, t)∇ · rh(x)dx−∫

∂K

c(ξ, t)n∂K · rh(ξ)dξ = 0 ∀rh ∈ (Vh)2.

(2.17)The numerical fluxes are defined as follows:

σ = σh − C11[[ch]]−C12[[σh]]

c = ch+ C12 · [[ch]]− C22[[σh]](2.18)

where C11, C12 and C22 are free parameters used to control the stability and accuracy of the scheme. Noticethat in (2.18) C11 and C22 are scalar quantities, while C12 is a vector.

Remark 2.3.1 Following [BR97a], the auxiliary variable is set equal to ∇c, and not to −a∇c as is usuallydone in mixed formulations.

Remark 2.3.2 In the DG method, equal order approximation is usually adopted for the primal and the aux-iliary variables ch and σh. This is different from what is done in mixed formulations, and simplifies theimplementation of the algorithm, since only the basis functions of Vh must be computed.

The semidiscrete problem represented by (2.17) and (2.18) is then integrated in time as in the hyperbolic case.

Typically, in the LDG scheme the parameter C22 is assumed to be zero, because this choice allows for a localcomputation of the auxiliary variable σh in terms of ch, which is essential when explicit time integrationschemes are adopted. When C22 6= 0, on the other hand, (2.17)2 constitutes a global system that must besolved at each time level to compute σh. The parameter C12 can be used to suitably vary the stencil of theLDG formulation, as shown in [CS98]. When C12 = 0, the method proposed by Bassi and Rebay in [BR97a]is recovered. Finally, C11 introduces a jump penalization and influences the accuracy of the scheme and, in thesteady state case, its stability.

The accuracy of the LDG scheme depends on both C11 and C22 in a rather involved way, as extensivelydiscussed in [CCPS00]. For k ≥ 1, it can be expected that the L2 norm of the error is proportional to hk+ 1

2 ,and the optimal convergence rate O(hk+1) can be recovered for appropriate values of C11 and C22. However,numerical experiments always show the optimal convergence rate O(hk+1), and it is possible that a similarphenomenon to what discussed in [Pet91] occurs for the parabolic problem as well. A particular treatment isnecessary for the case k = 0, where suitable values of the coefficients C11 and C22 are required to ensureconvergence of the numerical scheme. Indeed, the LDG scheme is one of the few methods for second orderparabolic problems that actually converges for piecewise constant approximations.

2.4. The Local Discontinuous Galerkin Method forNavier–Stokes Equations

In this section, we consider the application of the DG method to the complete Navier–Stokes equations, fol-lowing [BR97a] and [LK99]. We also mention the work by Dawson and Aizinger [DA05], where the LDG

21


formulation is applied to the shallow water equations. Since the problem presents a hyperbolic componenttogether with a parabolic component, the complete DG formulation can be obtained by combining the twoapproaches illustrated in Sect. 2.2 and Sect. 2.3.

We first introduce the following conservative compact form for the Navier–Stokes Equations in conservationform:

∂q∂t

+∇ · Fe(q) +∇ · Fv(q,∇q) = G(q), (2.19)

where

q =

ρVE

Fe(q) =

V

1ρV ⊗V + p I

1ρHV

Fv(q,∇q) =

0Fv

V(q,∇q)Fv

E(q,∇q)

G(q) =

0ρg0

.

All the above variables are defined in Sect. 1.3, and the viscous fluxes are defined as

FV = −µ[∇v +∇vT + λ∇ · v I

], FE = − γµ

Pr∇ei + v · FV,

where µ and λ are the two viscosity coefficients and Pr =ν

α=µcpk

denotes the Prandtl number, k and α

being the thermal conductivity and diffusivity, respectively. It follows thatγµ

Pr=

k

cvand, using the Stokes

hypothesis, we can set λ = −23

. Problem (2.19) can now be rewritten introducing the auxiliary variable S as∂q∂t

+∇ · Fe(q) +∇ · Fv(q,S) = G(q),

S −∇q = 0.(2.20)

An approximation (qh,Sh) = (qh(x, t),Sh(x, t)) to the solution (q(x, t),S(x, t)) of (2.20) is sought, suchthat (qh,Sh) ∈ (Vh)4 × (Vh)8 at each time level. In the following, for the sake of clarity we will oftenomit the dependence of qh and Sh on (x, t). Also, all products and differential operators are intended to beapplied separately to each density, momentum and energy component of their arguments. Multiplying the twoequations (2.20) by test functions vh ∈ (Vh)4 and Rh ∈ (Vh)8, respectively, integrating over K ∈ Th andreplacing the exact solution by its approximation, we get

d

dt

∫K

qhvhdx+∫

K

∇ · Fe(qh)vhdx+∫

K

∇ · Fv(qh,Sh)vhdx =∫

K

G(qh)vhdx, ∀vh ∈ (Vh)4∫K

Sh · Rhdx−∫

K

∇qh · Rhdx = 0, ∀Rh ∈ (Vh)8.

Then, formally integrating by parts and introducing the numerical fluxes he,K(qh), associated with Fe, andq(qh,Sh) and S(qh,Sh), associated with Fv , we obtain the following discrete problem:find (qh(·, t),Sh(·, t)) ∈ (Vh)4 × (Vh)8 such that for all t ∈ [0, T ] and ∀K ∈ Th

d

dt

∫K

qhvhdx−∫

K

Fe(qh) · ∇vhdx+∑

e∈∂K

∫e

he,K(qh)vhdξ

−∫

K

Fv(qh,Sh) · ∇vhdx+∫

∂K

Fv(q, S) · n∂Kvhdξ =∫

K


Sh · Rhdx+∫

K

qh∇ · Rhdx−∫

∂K

qn∂K · Rhdξ = 0, ∀Rh ∈ (Vh)8.

(2.21)

The definition of the numerical fluxes is now a critical element in the success of the DG formulation. Indeed, adifferent treatment is adopted for the hyperbolic and parabolic numerical fluxes.

22

2.4. The Local Discontinuous Galerkin Method for Navier–Stokes Equations

As for the hyperbolic component of the numerical flux, a generalization to nonlinear problems of the upwindflux (2.5) is required. This can be accomplished by taking advantage of the finite volume nature of the DGformulation, because, to construct he,K(qh) = he,K(qh(ξint(K), t),qh(ξext(K), t)), we can resort to the well–known approximate Riemann solvers developed for finite volume formulations. In order for the numericalproblem (2.21) to be well posed, he,K(q1,q2) must be a Lipschitz, consistent, monotone flux, that is, werequire that

• he,K(q,q) = Fe(q) · ne,∂K ,

• he,K(q1,q2) is nondecreasing in q1 and nonincreasing in q2,

• he,K(·, ·) is (globally) Lipschitz,

• he,K(·, ·) satisfies (2.6).

Possible choices for the hyperbolic numerical flux are (in the following, it is assumed that with each state q acomplete set of mechanical and thermodynamical variables v, ρ, . . . , is associated):

• the Godunov flux (see [God59]) he,K(q1,q2) = Fe(q 12(0)) · ne,∂K , where q 1

2(ξ/t) is the similarity

solution of the Riemann problem∂q∂t

+∇ · Fe(q) = 0

q(ξ, 0) =

q1 if ξ · ne,∂K < 0q2 if ξ · ne,∂K > 0

evaluated at ξ/t = 0;

• the Roe flux introduced in [Roe81]. A complete description of the flux can be found in [Tor97], while anexample of application to environmental flow problems is given in [DA05];

• the Rusanov flux (see [Rus61])

he,K(q1,q2) =12

[Fe(q1) · ne,∂K + Fe(q2) · ne,∂K − |λ| (q2 − q1)] , (2.22)

where |λ| = max |v1|+ a1, |v2|+ a2, with v1,2 = v1,2 · ne,∂K and a1,2 =√γRT1,2.

The viscous numerical fluxes are defined following (2.18). In particular, we will consider the Bassi and Rebaymethod, defined by

q = qh , S = Sh . (2.23)

The definition of the numerical fluxes completes the spatial discretization of the Navier–Stokes equations. Thetime integration can be carried out with a Runge–Kutta scheme, as discussed in Sect. 2.2. The implementationdetails of the resulting scheme can be found in [BR97a].

2.4.1. Strong form of the Local Discontinuous Galerkin method for theNavier–Stokes equations

As discussed in Sect. 2.2.2, it can be convenient to perform a counterintegration by parts on the weak formula-tion of the DG method. Concerning the Navier–Stokes equations, the following strong form will be considered

23


in the remainder of this thesis:find (qh(·, t),Sh(·, t)) ∈ (Vh)4 × (Vh)8 such that for all t ∈ [0, T ] and ∀K ∈ Th

d

dt

∫K

qhvhdx+∫

K

∇ · Fe(qh)vhdx−∑

e∈∂K

∫e

(Fe(qh) · n∂K − he,K(qh)) vhdξ

−∫

K

Fv(qh,Sh) · ∇vhdx+∫

∂K

Fv(q, S) · n∂Kvhdξ =∫

K


Sh · Rhdx−∫

K

∇qh · Rhdx+∫

∂K

(qh − q)n∂K · Rhdξ = 0, ∀Rh ∈ (Vh)8.

(2.24)Notice that, in order to avoid the evaluation of the derivatives of the viscous flux, which would represent arather involved computation, the fourth term in (2.24)1 is not counterintegrated by parts.

24

Chapter 3.

The Semi–Lagrangian DiscontinuousGalerkin Method

In this chapter the semi–Lagrangian and the Discontinuous Galerkin formulations arecombined to devise a novel discretization scheme for the approximation of the LinearAdvection Equation, namely, the semi–Lagrangian Discontinuous Galerkin formulation.Exploiting the formalisms introduced in Chapts. 1 and 2, the new formulation is describedin Sect. 3.2. Section 3.3 presents the Von Neumann stability analysis of the novel for-mulation for the one–dimensional case. A monotonization algorithm based on the FluxCorrected Transport approach is then illustrated in Sect. 3.4. Finally, an extensive nu-merical validation of the proposed scheme is carried out in Sect. 3.5, considering someclassical one–dimensional and two–dimensional test cases.

3.1. Introduction

The purpose of the present chapter, which in essence recovers the material presented in [RBS06], is to introducea flux form SL discretization for the scalar advection equation that employs a DG formulation to reconstructthe numerical solution within each control volume. We will refer to this novel discretization technique asthe semi–Lagrangian Discontinuous Galerkin (SLDG) approach. In doing so, the proposed method aims atcombining the accuracy and locality of the DG method with the computational efficiency and robustness ofsemi–Lagrangian techniques. On one hand, the use of SL backward trajectories allows to achieve unconditionalstability, irrespective of the value of the Courant number, thus overcoming the severe stability restrictions (2.9)enforced by the DG formulation. On the other hand, the potential loss of accuracy of standard SL methods atlow Courant numbers pointed out in [FF98] does not affect the SLDG scheme, as demonstrated by a number ofnumerical experiments. Furthermore, in the case of large systems of advection–diffusion–reaction equations,as typical of environmental modelling applications [HWM+03, Tim01], the extra effort needed to computethe trajectories is required only once for the whole system, so that the potential overhead associated with thiseffort becomes negligible, as demonstrated in [LR05]. Finally, the proposed method is characterized by acomputational stencil that is similar to those of standard Eulerian DG formulations. This means that high–order approximations can be constructed locally, without involving a large number of neighbouring elements,thus making domain decomposition–based parallelization approaches more straightforward. As it will be clearfrom the description of the numerical method, the use of an elementwise variable–degree formulation is alsopossible, leaving unaltered the global and local mass conservation properties of the scheme. This feature of

Chapter 3. The Semi–Lagrangian Discontinuous Galerkin Method

the formulation will be exploited in a future improved implementation of the scheme to reduce computationalcosts.

The novel formulation is presented here considering the LAE (2.1) and assuming that the advective flow u isincompressible. In the case where diffusive phenomena are included in the mathematical model, we adopt anoperator splitting approach combined with the classical DG formulations described in Sect. 2.3. The operatorsplitting technique addressed in the present chapter should be regarded as a preliminary attempt to incorporatediffusive terms in the SLDG formulation, and possible alternatives to treat diffusion in conjunction with theSLDG method will be the subject of future investigations.

A brief outline of the chapter is as follows. The SLDG method is described in Sect. 3.2. A von Neumannstability analysis for the constant coefficient, one dimensional case is then carried out in Sect. 3.3, showing thatthe method is stable for an arbitrary value of the Courant number. Since the SLDG is not inherently monotonicin its higher order version, a monotonization approach based on the Flux Corrected Transport (FCT) techniqueis introduced and discussed in Sect. 3.4, while in Sect. 3.5 the interesting properties of the new method aredemonstrated by a number of numerical tests relevant in advection dominated flows. Future developments andpossible applications are discussed in the concluding Sect. 3.6.

3.2. The Semi-Lagrangian Discontinuous Galerkin Method

In this section we describe in detail the SLDG method. In doing this, we combine the unified framework forgeneralized Godunov methods proposed in [Mor98] (see Sect. 1.1.1) with the DG finite element formulationintroduced and analyzed in [CHS90, CS98] in the case of nonlinear hyperbolic problems (see Chapt. 2).

3.2.1. Preliminaries

Let Ω be an open bounded domain of R2, with boundary ∂Ω ≡ Γ and outward unit normal vector nΓ, wherea solution of (2.1) is to be approximated, and let Th denote a partition of Ω into Nel triangular elements K.This latter choice allows an easy generation of approximation basis functions of arbitrary polynomial order andprovides a great flexibility to the geometrical discretization of the computational domain. The application ofthe scheme to include quadrilateral elements and to treat three-dimensional problems can be carried out alongthe same lines as in [Gir00a, Gir00b]. The area of an element K is denoted by |K|, while its boundary andoutward unit normal vector are ∂K and n∂K , respectively. The diameter of K is hK and h = max

K∈Th

hK . The

set of all the edges of K is EK , while Eh is the set of the Nedges edges of the triangulation, |e| denoting thelength of a generic edge e ∈ Eh. For each internal edge e ∈ Eh, a normal unit vector ne is arbitrarily fixed,and K1

e , K2e denote the two elements of Th sharing the edge e and such that ne is directed from K1

e to K2e . If

e ∈ Γ, then ne ≡ nΓ and K1e is the element of Th having e as a boundary edge. Consistently, let σK1

e ,e = 1and σK2

e ,e = −1, in such a way that σKie,ene is the outer normal unit vector associated with edge e of element

Kie, for i = 1, 2.

Equation (2.1) is to be approximated in Ω supplied with inflow boundary conditions and under the assumptionthat the given velocity vector field u is solenoidal, i.e., ∇ · u = 0. In view of the numerical approximation ofeq. (2.1), we need to introduce the projection of u over the finite element space of Raviart–Thomas of lowestdegree [RT77]. With a slight abuse of notation, we indicate throughout the chapter this projection by the symbolu. The numerical importance of using this projected field relies on the fact that i) it allows an automatic fluxconservation across interelement boundaries and ii) it is easy to check that u is piecewise constant on eachK ∈ Th. The construction of u only requires for each edge e ∈ Eh the constant quantity ue, that is the normal

26

3.2. The Semi-Lagrangian Discontinuous Galerkin Method

velocity component in the direction of ne. Then, for each time interval [tn, tn+1], with ∆t = tn+1 − tn,the normal components of the velocity field to the edges of Th are assumed to be given at the intermediatetime level tn+ 1

2 and are denoted by un+ 12

e , e = 1, . . . , Nedges, so that the (discrete) divergence–free constraintamounts to requiring that ∑

e∈EK

σK,eun+ 1

2e |e| = 0 ∀K ∈ Th. (3.1)

The time dependence of u and ue will often be omitted in the following for the sake of simplicity. We notice thatassuming that u can be completely determined by its normal fluxes eases the future coupling of the proposedscheme to mass conservative methods for environmental flows, such as those proposed in [BKHR05, BR05,CW00, MQS99]. Moreover, assuming a velocity field piecewise constant in time, although limiting a priori theformal accuracy of the method, corresponds to what is actually computationally feasible when coupling traceradvection to most semi–Lagrangian models for fluid flow.

3.2.2. Spatial discretization

The spatial discretization of (2.1) is carried out initially along the usual lines of DG methods and problem (2.3)is obtained. At this point, we depart from the usual DG technique and follow the path of generalized Godunovmethods as presented and analyzed in [Mor98]. With this aim, we integrate (2.3) in time between tn and tn+1,to obtain the following weak form of the linear advection equation∫

K

ch(x, tn+1) vh(x) dx =∫

K

ch(x, tn) vh(x) dx

+∫ tn+1

tn

dτ

∫K

ch(x, τ)u(x, τ) · ∇vh(x) dx

−∫ tn+1

tn

dτ

∫∂K

ch(ξ, τ)u(ξ, τ) · n∂K vh(ξ) dξ ∀vh ∈ Vh.

(3.2)

For each element K ∈ Th, the discrete degrees of freedom associated with the numerical solution at a giventime level tn are denoted by cnj,Kj∈Jk

, with Jk = 0, . . . , Nk − 1, so that an approximate numericalsolution can be reconstructed locally for all K ∈ Th as

ch(x, tn)|K =∑j∈Jk

cnj,Kφj,K(x). (3.3)

In the following, the functions φj,K(x) are taken to be an orthogonal basis for Pk(K) such that∫K

φi,K(x)φj,K(x)dx = |K|δij

(and, in particular, φ0,K(x) = 1K(x), 1K being the characteristic function associated with element K, and∫Kφj,K(x)dx = 0 for j > 0). For brevity, we set henceforth cnh(x) = ch(x, tn).

3.2.3. Time discretization

The next step is to derive from (3.2) a full space–time discretization. This can be done by exploiting theevolution operator E(t, s) introduced in Sect. 1.1.1. More precisely, following the ideas proposed in [LLM96,LR96], we can use (1.5) and the approximate evolution operator En

s , also introduced in Sect. 1.1.1, to evaluate

27


the right hand side of (3.2), so that the SLDG method can be defined for each element K ∈ Th by

|K| cn+1j,K = |K| cnj,K +

∫ ∆t

0

ds

∫K

[Ens c

nh](x)u(x, tn+ 1

2 ) · ∇φj,K(x) dx (3.4)

−∫ ∆t

0

ds∑

e∈EK

∫e

[Ens c

nh](ξ)un+ 1

2e φj,K(ξ) dξ, ∀j ∈ Jk.

For ease of notation, the time dependency of the velocity field will be omitted in the remainder.

3.2.4. The fully discrete SLDG approximation

In order to obtain a fully discrete method, the integrals in space and time in (3.4) must be replaced by appropri-ate quadrature rules. In the present implementation of the proposed method, Gaussian quadrature rules are usedfor the integration in space. Gaussian points xvLe

v=1, yvLf

v=1, are introduced on the edges and the elements,respectively. The corresponding Gaussian weights are denoted by ωvLe

v=1, ˜ωvLf

v=1. For the integration intime, a simple composite rule is applied in the present implementation. For each element K and for each edgee, we define intermediate time levels sK

mM(K)m=0 , se

mM(e)m=0 . For convenience, the dependency on the edge and

element is often dropped and should be recovered from the context. The intermediate time steps are such thats0 = 0, sM = ∆t and ∆sm = sm − sm−1. Formally, we make the approximation

∫ ∆t

0

Ens ds ≈

M−1∑m=0

Ens

m+ 12

∆sm, (3.5)

where now sm+ 12

= sm+∆sm

2 .More accurate composite integration rules can of course be used along the same

lines. The numerical trajectories X(x, tn + s; tn) necessary for the complete definition of Ens are computed by

a simple backward Euler method with time substeps given by the quantities ∆sm. Given these definitions, thefully discrete SLDG approximation of equation (2.1) can then be defined for each K ∈ Th as

|K|cn+1j,K = |K|cnj,K

+M−1∑m=0

Lf∑v=1

[Ens

m+ 12

cnh](yv)u(yv) · ∇φj,K(yv) ∆sm˜ωv (3.6)

−∑

e∈EK

σK,eue |e|M−1∑m=0

Le∑v=1

[Ens

m+ 12

cnh](xv)φj,K(xv) ∆sm ωv, ∀j ∈ Jk.

It must be remarked that the approximation (3.5) of the evolution operator eliminates the ambiguity in thedefinition of the numerical fluxes along interelement boundaries, since in all cases with non zero advectingvelocity the quantity [En

sm+ 1

2

cnh](xv) is uniquely defined for m = 0, · · · ,M − 1. In the special case where

piecewise constant finite elements are considered, the following finite volume method is recovered from (3.6)

|K| cn+10,K = |K| cn0,K −

∑e∈EK

σK,eue |e|M−1∑m=0

Le∑v=1

[Ens

m+ 12

cnh](xv) ∆sm ωv, (3.7)

where the quantity cn0,K is the discrete degree of freedom representing the average of the concentration overelement K ∈ Th and cnh is a piecewise constant function over Th.

28

3.3. Linear stability analysis in the one-dimensional case

3.3. Linear stability analysis in the one-dimensional case

In this section, we carry out the stability analysis of the SLDG scheme in the von Neumann sense, alongthe lines of Sect. 1.1.1 and of [CC87]. The proof will be given only for the case of linear polynomial ap-proximations P1, although it maintains its validity for higher–order elements. With this aim, let us considerproblem (1.10), and let Th be a uniform triangulation of [0 , L], with h denoting the (uniform) amplitude ofeach element K ∈ Th and ∆t denoting the time step. Finally, for each Ki ∈ Th, let xi, xi+ 1

2and xi− 1

2be the

midpoint and the points on the boundary of Ki, respectively (see also Fig. 3.1). The key stability parameter isthe Courant number C = u∆t

h , which can be split in its integer and fractional part

C = m+ γ, m ∈ N, γ ∈ [0 , 1).

Introducing the cell characteristic time τ = hu we have also ∆t = (m + γ)τ . Denoting now by aj and bj the

h

xi−1

2x

i+12

xixi−1xi−m−1

umτuγτ

ξξi−1

2

ξi+1

2

u∆t

Figure 3.1.: Piecewise linear solution of the advection equation in the one–dimensional case. Notice also thegeometrical representation of the integer and fractional parts m and γ of the Courant number.

degrees of freedom of ch with respect to the P1 hierarchical basis, we have

cnh(x)|Kj= an

j + 6 bnj (x− xj)/(√

3h) ∀Kj ∈ Th.

The SLDG formulation (3.6) for the evolution of ch from time level tn to tn+1 can then be expressed as afunction of aj and bj as

an+1j = an

j−m − γ(anj−m − an

j−m−1)−√

3γ(1− γ)(bnj−m − bnj−m−1),

bn+1j = bnj−m +

√3γ(1− γ)(an

j−m − anj−m−1)

− 3γ(1− γ)(bnj−m + bnj−m−1)− 2γ2(

32 − γ

)(bnj−m − bnj−m−1).

(3.8)

Proposition 3.3.1 The SLDG method (3.8) is unconditionally L2-stable.

Proof. We have||cn

h||2L2(0,L) = hX

j

(anj )2 + (bn

j )2. (3.9)

29


We can associate with chn the piecewise constant vector function [an

h(x), bnh(x)]T with an

h(x) =P

j anj 1Kj (x) and

bnh(x) =

Pj bn

j 1Kj (x), so that (3.9) can be rewritten as

||cnh||2L2(0,L) = ||an

h||2L2(0,L) + ||bnh||2L2(0,L). (3.10)

Let us consider the Fourier series anh(x) =

Pk∈Z An

kei 2kπL

x and bnh(x) =

Pk∈Z Bn

k ei 2kπL

x associated with anh and bn

h ,where i =

√−1. Applying the Bessel-Parseval equality to (3.10) yields

||cnh||2L2(0,L) = L

Xk∈Z

(|Ank |2 + |Bn

k |2). (3.11)

The coefficients An+1k , Bn+1

k of the SLDG method can be expressed as linear combinations of Ank , Bn

k as follows

ˆAn+1

k , Bn+1k

˜T= e−iθmG(θ, γ) [An

k , Bnk ]T , (3.12)

where θ = 2kπhL

and G is the skew–symmetric amplification matrix with entries

G11(θ, γ) = 1− γ(1− e−iθ), G12(θ, γ) = −√

3γ(1− γ)(1− e−iθ)

G22(θ, γ) = 1− 3γ(1− γ)(1 + e−iθ)− γ2(3− 2γ)(1− e−iθ).

Let λ1,2 denote the eigenvalues of G, with |λ2| ≤ |λ1|. It can be checked that λ1 6= λ2 for all θ ∈ [−π , π], and that forθ → 0

|λ1| = 1− 1

72γ(1− γ)(1− γ + γ2)θ4 +O(θ6),

so that, ∀θ ∈ [−π , π] we have |λ1| < 1 for |γ| ≤ 1, as can be seen from the plots of λ1, λ2 in Fig. 3.3. As a consequence,the following inequality holds

||cn+1h ||2L2(0,L) < ||cn

h||2L2(0,L)

which proves the unconditional stability of the SLDG method in the von Neumann sense.

−1 −0.5 0 0.5 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

−1 −0.5 0 0.5 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Figure 3.2.: Plot of λ1 (red) and λ2 (blue) for γ = 0.05, 0.2, 0.4 (left) and γ = 0.6, 0.8, 0.9 (right). Green:unit circle.

We notice that the stability result proved in this section is analogous to what is usually obtained for standardSL methods (see, e.g., [BM82, FF98]), and it provides an improvement of the stability analysis carried outin [CS91], which yields the stability constraint γ < 1

3 for the DG method using linear finite elements (see (2.9)with k = 1).

30

3.4. Monotonicity

3.4. Monotonicity

In this section, we propose and illustrate a technique to turn the SLDG discretization (3.6) into a monotonicmethod. In order to achieve this important property, we first show in Sect. 3.4.2 that the piecewise constant partof the approximate solution satisfies a discrete maximum principle, that is

minK∈Th

cn0,K ≤ cn+10,K ≤ max

K∈Th

cn0,K ∀K ∈ Th. (3.13)

Then, we apply to the SLDG formulation the well–known Flux Corrected Transport (FCT) technique [Zal79] toenforce monotonicity of the higher order approximate solution. It is important to notice that, as in the contextof DG methods for scalar conservation laws, the proof of a discrete maximum principle is divided into twosteps [CHS90]. In the first step, a discrete maximum principle is proved for the degrees of freedom representingthe average of the solution over each element K (i.e., in our case, the values cn+1

0,K ). Then, a discrete maximumprinciple is established for the DG method using higher order elements by limiting the slopes of the numericalsolution, represented by the degrees of freedom cnj,K , with j = 1, . . . , Nk−1, in (3.3). In the present approach,instead, we enforce monotonicity by a suitable correction of the edge flux contributions in (3.4) using the FCTapproach, as explained in Sect. 3.4.3.

3.4.1. Definition of edge fluxes

In this preliminary section, we introduce edge fluxes into the SLDG formulation (3.6). For each edge e ∈ ∂K,we define the approximate flux associated with the j−th degree of freedom as

F je = ue |e|

M−1∑m=0

Le∑v=1

[Ens

m+ 12

cnh,j ](xv) ∆sm ωv, ∀j ∈ Jk,

where cnh,j(x) =∑

K∈Thcnj,Kφj,K(x) represents the j−th component of cnh(x). Using the above definition,

the update (3.7) for the mean value of ch can be written in the following equivalent form

cn+10,K = cn0,K −

∑e∈EK

σK,eue |e||K|

M−1∑m=0

Le∑v=1

[Ens

m+ 12

cnh](xv) ∆sm ωv

= cn0,K −∑

e∈EK

σK,e

|K|F 0

e −Nk−1∑j=1

∑e∈EK

σK,e

|K|F j

e .

(3.14)

It is now convenient to redefine F 0e as

F 0e =

∫ ∆t

0

ds

∫e

[Ens c

nh,0](ξ) dξ =

∫e

dξ

∫ ∆t

0

[Ens c

nh,0](ξ)ds. (3.15)

Due to the definition of Ens , one has∫ ∆t

0

[Ens c

nh,0](ξ) ds =

∫ ∆t

0

cnh,0(X(ξ, tn + s; tn)) ds.

It must be remarked that, since the time dependency of the velocity field is assumed to be frozen during eachtime step, the set of points X(ξ, tn + s; tn), s ∈ [0 , ∆t], coincides with those spanned by the backwardtrajectory X(ξ, tn+1; tn+1 − s), s ∈ [0 , ∆t]. Noting that cnh,0 is piecewise constant over Th, the previousrelation can be written as ∫ ∆t

0

cnh,0(X(ξ, tn + s; tn)) ds =∑K′

cn0,K′∆sξK′ ,

31


where ∆sξK′ denotes the amount of time during which X(ξ, tn+1; tn+1 − s) ∈ K ′ and the sum is extended

over all elements crossed by X(ξ, tn+1; tn+1 − s). As a result,∑

K′ ∆sξK′ = ∆t independently of ξ and

F 0e = ue|e|∆t

∑K′∈Te

αK′,ecn0,K′ , (3.16)

where we set

αK′,e =1

|e|∆t

∫e

∆sξK′ dξ (3.17)

and where Te is the set of all elements crossed by X(ξ, tn+1; tn+1 − s) for any ξ ∈ e. Noting that for eachedge e ∈ EK we have ∑

K′

∫e

∆sξK′ dξ = |e|∆t,

from definition (3.17) it turns out that αK′,e ≥ 0 and that we also have∑K′∈Te

αK′,e = 1. (3.18)

It is important to notice that, in the case of a velocity field defined as in Sect. 3.2.1 and satisfying exactly thediscrete divergence–free constraint (3.1), the quantities ∆sξ

K′ and αK′,e can be computed exactly.

3.4.2. Monotonicity of ch,0

In this section, we carry out the first step of the monotonicity proof by showing that the mean value of ch,defined as

cn+10,K = cn0,K −

∑e∈EK

σK,e

|K|F 0

e , (3.19)

satisfies the discrete maximum principle.

Proposition 3.4.1 The quantity cn+10,K satisfies the discrete maximum principle (3.13).

Proof. Let us show that the update cn+10,K can be expressed as a linear combination of cn

0,K′ with nonnegative coefficients.With this aim, we introduce the following sets

E +K = e ∈ EK : σK,eue ≥ 0, E−

K = e ∈ EK : σK,eue < 0,

which represent the outflow and inflow boundaries of K, respectively. We also introduce the two following subsets of Te

T +K =

[e∈E+

K

Te, T −K =

[e∈E−

K

Te.

Clearly, EK = E +K ∪ E−

K , which can be used to reformulate (3.19) as

cn+10,K = cn

0,K −X

K′∈T +K

Xe∈E+

K

|ue||e|∆t

|K| αK′,ecn0,K′ +

XK′∈T −

K

Xe∈E−

K

|ue||e|∆t

|K| αK′,ecn0,K′ . (3.20)

For simplicity, we make now the assumption that K /∈ T −K , which excludes the case of very high Courant numbers or

flows with very strong deformation (i.e., high Lipschitz numbers according to the definition in [SP92]). The proof can

32

3.4. Monotonicity

be extended to cover also these cases, but it is far simpler under the above assumption. In any case, irrespective of thisrestriction, one has T +

K = K ∪ (T +K ∩ T −

K ). Relation (3.20) can then be rewritten as

cn+10,K = αn

K cn0,K +

XK′∈(T −

K∩T +

K)

βnK′ cn

0,K′ +X

K′∈(T −K\T +

K)

γnK′ cn

0,K′ ∀K ∈ Th, (3.21)

where

αnK =

0B@1− ∆t

|K|X

e∈E+K

|ue||e|αK,e

1CA ,

βnK′ =

∆t

|K|

0B@ Xe∈E−

K

|ue||e|αK′,e −X

e∈E+K

|ue||e|αK′,e

1CA ,

γnK′ =

∆t

|K|X

e∈E−K

|ue||e|αK′,e.

Let us now show that αnK , βn

K′ and γnK′ are nonnegative and that

αnK +

XK′∈(T −

K∩T +

K)

βnK′ +

XK′∈(T −

K\T +

K)

γnK′ = 1. (3.22)

To prove this latter property, let us first set cn0,J = 1 for all elements J = K, K′ at the right hand side of (3.21). Then, (3.22)

immediately follows using (3.18) and noting that

∆t

|K|X

e∈EK

σK,e|ue||e| =∆t

|K|

264 Xe∈E−

K

|ue||e| −X

e∈E+K

|ue||e|

375 = 0.

Let us now check that the coefficients in (3.21) are nonnegative. This is immediate for γnK′ . For αn

K , nonnegativity canbe proved by using the definition of sξ

K′ and the properties of the Raviart–Thomas projection u discussed in Sect. 3.2.1.For βn

K′ , nonnegativity is ensured by the fact that, for an approximation of the characteristic lines based on the piecewiseconstant Raviart–Thomas projection u, one has αK′,e′ ≥ αK′,e′′ if e′ ∈ E−

K , e′′ ∈ E +K . Then, for each K′ ∈ (T −

K ∩T +K ),

we have

Xe′∈E−

K

|ue′ ||e′|αK′,e′ −X

e′′∈E+K

|u′′e ||e′′|αK′,e′′ ≥ mine′∈E−

K

αK′,e′

264 Xe∈E−

K

|ue||e| −X

e∈E+K

|ue||e|

375 .

Using again the fact that u satisfies (3.1), we immediately get that also βnK′ ≥ 0, which concludes the proof.

3.4.3. Monotonicity of ch,1 and ch,2

In this section, we carry out the second step of the proof by deriving a monotonic higher order method throughthe use of the Flux Corrected Transport (FCT) technique (see [Zal79]). In order to apply the FCT framework,we identify in (3.14) the quantities F 0

e and Ae =∑Nk−1

j=1 F je as low order diffusive flux and antidiffusive flux,

respectively. Thus, appropriate limiting coefficients Ce ∈ [0, 1] can be introduced for the antidiffusive fluxesin such a way to obtain the monotonized higher order scheme

cn+10,K = cn0,K −

∑e∈EK

σK,e

|K|F 0

e −∑

e∈EK

σK,e

|K|CeAe ∀K ∈ Th. (3.23)

The computation of the coefficients Ce is done along the same lines as in [Zal79], and the algorithm is summa-rized here for sake of completeness.

For each K ∈ Th:

33


• compute the low order solution cn+10,K using (3.23) with Ae = 0;

• compute the maximum and minumum allowable mean values cmax0,K and cmin

0,K from the upstream neigh-boring elements;

• compute P+K and P−K as the sum of all antidiffusive fluxes into and away from K, respectively;

• computeQ+

K = (cmax0,K − cn+1

0,K )|K|, Q−K = (cn+1

0,K − cmin0,K )|K|;

• set

R+K =

min(1, Q+

K/P+K ) if P+

K > 00 if P+

K = 0, R−K =

min(1, Q−

K/P−K ) if P−K > 0

0 if P−K = 0.

Finally, for each edge e ∈ Eh, set

Ce =

min(R+

K2e, R−K1

e) if Ae ≥ 0

min(R−K2e, R+

K1e) if Ae < 0.

3.5. Numerical Results

In this section, we demonstrate the accuracy and stability of the SLDG method on one and two–dimensionalbenchmark test cases for passive tracer advection. Tests are performed for the plain, non monotonic (NM) andthe FCT monotonized (M) versions of the SLDG method. A possibile approach, based on a DiscontinuousGalerkin formulation, is also proposed to include in SLDG the discretization of a diffusive term.

For the one–dimensional case, advection of both continuous and discontinuous profiles is considered. In orderto compare the accuracy of the proposed SLDG scheme to that of well–known reference schemes, the sametest problems as in [HO87] are considered. Results indicate that the NM version of SLDG method is secondand third order accurate when P1 and P2 elements are used, respectively. The M version of the scheme retainsits accuracy away from local extrema (as usual for monotonic schemes); moreover, the absolute error valuescompare well with those reported in the reference.

For the two–dimensional case, we study solid body rotation and deformational flow tests for which analyticsolutions are available. The results obtained using the SLDG method are then compared with those providedby its parent methods, e.g., the standard non conservative semi–Lagrangian (SL) method and the Runge–KuttaDiscontinuous Galerkin (RKDG) method. For the SL and SLDG runs the backward trajectories are approxi-mated by using the backward Euler method with substepping (see, e.g., [Gir99, RBC05]). These comparisonshighlight several attractive properties of the SLDG formulation, which appears to merge effectively the SL andDG methods without any loss in accuracy. The FCT based monotonization approach described in Sect. 3.4appears to be superior to the slope limiting approach proposed in [CHS90] and does not exhibit the excessivesharpening of smooth profiles reported e.g. in [SS96] for more traditional applications of FCT. In particular, theidea of retaining higher order degrees of freedom in the computation of the monotonized flux for the piecewiseconstant component of ch seems to be highly beneficial to the overall quality of the computed approximatesolution, at least in the case of P1 elements.

In all the tests, the error norms

L2 error =||c− ch||L2(Ω)

||c||L2(Ω), L∞ error =

||c− ch||L∞(Ω)

||c||L∞(Ω)

34


are computed, where c denotes the exact solution. We also compute the conservation error∫Ω(c− ch) dx and

the two following error measures (see [BS92] and [Tak85]):

Dissipation error = [σ(c)− σ(ch)]2 + (c− ch)2

Dispersion error = 2[σ(c)σ(ch)− 1

|Ω|

∫Ω

(c− c)(ch − ch) dx],

where, for a function φ, we set

φ =1|Ω|

∫Ω

φdx, σ(φ) =

√1|Ω|

∫Ω

(φ− φ)2 dx.

3.5.1. One–Dimensional Advection

In this section, we consider problem (1.10) with u = 1 on the space–time domain [−1 , 1]× [0 , 2]. In applyingSLDG method to this problem, backward trajectories and flux integrals are evaluated exactly. For ease ofcomparison with the results reported in [HO87], the L∞ and L1 absolute error norms are computed in thiscase, as well as the maximum and minimum values of the approximate solution. For the first test we setc0(x) = sin(πx) and consider P1 and P2 elements on several computational grids with an increasing numberof elements Nel and a constant Courant number C = 0.8. Tab. 3.1 shows the computed error norms for boththe NM and M versions of SLDG with P1 elements. It can be checked that, in absence of monotonization, theproposed scheme is second order accurate in both L∞ and L1 norms, while introducing the monotonizationthe full second order accuracy can only be retained in the L1 norm. As for the absolute values of the error,

L∞ error L1 errorNel NM M NM M10 5.28 e− 02 1.10 e− 01 4.74 e− 02 6.23 e− 0220 1.31 e− 02 4.13 e− 02 1.16 e− 02 1.73 e− 0240 3.25 e− 03 1.34 e− 02 2.84 e− 03 4.12 e− 0380 8.06 e− 04 3.75 e− 03 7.02 e− 04 9.63 e− 04

160 2.01 e− 04 1.10 e− 03 1.75 e− 04 2.25 e− 04320 5.01 e− 05 3.10 e− 04 4.35 e− 05 5.36 e− 05p 2.01 1.69 2.02 2.04

Table 3.1.: Computed L∞ and L1 error norms and estimated convergence rate p for both the NM and M ver-sions of SLDG, with P1 elements and regular initial datum.

we observe that, with respect to the results reported in [HO87], the limited SLDG method lies in between thereference TVD scheme and the essentially non oscillatory “UNO2” scheme. The monotonic computed solutionand the exact solution in the case Nel = 20 are shown in Fig. 3.3, where it can be seen that local extrema aremostly responsible for the loss of second order accuracy. Tab. 3.2 shows the computed error norms for both theNM and M versions of SLDG with P2 elements. In absence of monotonization, the novel scheme appears tobe third order accurate, while the introduction of the FCT monotonization reduces the convergence rate. Thisreduction of the convergence rate is due to errors localized at local extrema, as can be checked from the thirdand sixth columns of Tab. 3.2, where the two intervals [−0.6, −0.4] and [0.4, 0.6] are not considered in theerror evaluation.

As second and third test cases, we consider discontinuous initial data, namely a square wave profile and the

35


−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 3.3.: One–dimensional advection test case with sinusoidal profile, C = 0.8 and Nel = 20. Solid line:piecewise linear SLDG solution; circles: monotonic mean values; dashed line: analytic solution.

L∞ error L1 errorNel NM M M∗ NM M M∗

10 3.35 e− 03 9.06 e− 02 5.98 e− 02 9.95 e− 04 5.80 e− 02 3.16 e− 0220 4.07 e− 04 4.17 e− 02 2.76 e− 02 1.13 e− 04 1.51 e− 02 7.17 e− 0340 5.13 e− 05 1.16 e− 02 4.08 e− 03 1.39 e− 05 2.93 e− 03 2.39 e− 0480 6.43 e− 06 3.28 e− 03 1.09 e− 04 1.73 e− 06 4.96 e− 04 4.71 e− 06160 8.05 e− 07 1.13 e− 03 1.21 e− 06 2.17 e− 07 9.06 e− 05 2.14 e− 07320 1.01 e− 07 5.68 e− 04 1.01 e− 07 2.71 e− 08 2.34 e− 05 2.57 e− 08p 2.99 0.99 3.58 3.00 1.95 3.06

Table 3.2.: Computed L∞ and L1 error norms and estimated convergence rate p for both the NM and M ver-sions of SLDG, with P2 elements and regular initial datum. Columns indicated by M∗ refer to themonotonic SLDG scheme where the error is evaluated away from local extrema.

following irregular profile

c0(x) =

−x sin(3πx2/2) in [−1 , −1/3)| sin(2πx)| in [−1/3 , 1/3]2x− 1− 1

6 sin(3πx) in (1/3 , 1].

Monotonic solutions computed on grids with 20 and 40 elements and with C = 0.8 are shown in Fig. 3.4, whilethe corresponding errors are summarized in Tab. 3.3. It can be seen that the piecewise constant component ch,0

of the monotonic solution is bounded by the extreme values of the initial datum (while the complete solution isnot). The results compare well with those reported in [HO87] for the “UNO2” scheme.

3.5.2. Two–Dimensional Advection: Solid Body rotation

For the solid body rotation test, a stationary velocity field is considered, representing a rotating flow with fre-quency ω = 2π/1000 s−1 = 6.2832 · 10−3 s−1 around the point (1, 1) on the spatial domain Ω = (0, 2)2. Theinitial datum is either a compactly supportedC3 function with the shape of a cosine hill or a piecewise constant,

36


−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−1 −0.5 0 0.5 1 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Figure 3.4.: Left: square wave one–dimensional advection test case with C = 0.8 and Nel = 20: monotonicSLDG solution (solid line, mean values represented by circles), and analytic solution (dashed line).Right: irregular profile one–dimensional advection test case with C = 0.8 and Nel = 40: mono-tonic SLDG solution (solid line, mean values represented by circles), and analytic solution (dashedline).

square wave irregular profileL∞ error 6.98 e− 01 9.42 e− 01L1 error 1.29 e− 01 2.16 e− 01max(ch) 1 + 3.77 e− 02 1− 4.74 e− 02min(ch) −3.77 e− 02 −1 + 4.69 e− 01

max(ch,0) 1− 4.20 e− 05 1− 8.42 e− 02min(ch,0) 4.20 e− 05 −1 + 5.28 e− 01

Table 3.3.: Error norms for the one–dimensional tests with discontinuous initial data.

discontinuous function with the same support, while errors are evaluated at T = 4000 s, corresponding to 4full rotations.

In order to test the accuracy of the proposed SLDG method, we consider the standard test case of solid bodyrotation with smooth initial datum. Four unstructured computational grids of varying amplitude h are used,keeping the Courant number constant and equal to C = 0.25. Numerical quadratures are performed by settingLe = 2, Lf = 3 and M = 2 in (3.6). The characteristics of the computational grids are summarized in Tab. 3.4while the numerical results are shown in Tab. 3.5 for the non monotonic SLDG scheme.

h Nel dofs ∆th = 0.1 1372 4116 1.25h = 0.05 5458 16374 0.625h = 0.033 12222 36666 0.416h = 0.025 22172 66516 0.312

Table 3.4.: Computational grids for the convergence test.

The experimental average convergence rates derived from Tab. 3.5 are 2.00, 2.03 in the L2, L∞ norms, respec-tively. Analogous results have been obtained in the L1 norm. These results appear to be compatible with the

37


Rel. L2 Rel. L∞ Dissipation Dispersion Cons.error error error error error

h=0.1 3.53 e− 01 4.07 e− 01 1.96 e− 02 7.77 e− 02 −2.54 e− 16h=0.05 1.24 e− 01 1.49 e− 01 1.51 e− 03 1.11 e− 02 −3.92 e− 16h=0.033 5.58 e− 02 6.82 e− 02 1.96 e− 04 2.39 e− 03 1.81 e− 15h=0.025 2.73 e− 02 3.18 e− 02 3.36 e− 05 5.87 e− 04 2.63 e− 15

min(ch) max(ch) min(ch,0) max(ch,0)h=0.1 −4.51 e− 01 10− 3.61 e+ 00 −4.26 e− 01 10− 4.08 e+ 00h=0.05 −2.73 e− 01 10− 1.05 e+ 00 −2.50 e− 01 10− 1.42 e+ 00h=0.033 −1.21 e− 01 10− 4.18 e− 01 −1.11 e− 01 10− 5.72 e− 01h=0.025 −5.37 e− 02 10− 1.34 e− 01 −5.04 e− 02 10− 2.55 e− 01

Table 3.5.: Convergence test for the NM version of SLDG, smooth profile, C = 0.25.

supraconvergence estimates presented in [MS95].

It is interesting to compare the results summarized in Tab. 3.5 with the analogous ones obtained using theSL and RKDG formulations under the same working conditions (C = 0.25). Tab. 3.6 refers to the solutioncomputed by the SL method on three computational grids with P2 reconstruction, h = 0.1, 0.05, 0.025 andnumber of degrees of freedom Ndofs = 2825, 11077, 44665. Tab. 3.7 refers to the solution computed bythe RKDG method without slope limiting on two computational grids with P1 finite elements, h = 0.1, 0.05and Ndofs = 4116, 16374.

Rel. L2 Rel. L∞ Dissipation Dispersion Minerror error error error

h=0.1 7.42 e− 01 7.13 e− 01 1.02 e− 01 1.74 e+ 00 −7.00 e− 01h=0.05 5.18 e− 01 4.86 e− 01 1.77 e− 02 8.80 e− 01 −1.08 e+ 00h=0.025 2.29 e− 01 2.14 e− 01 1.16 e− 03 1.75 e− 01 −6.87 e− 01

Table 3.6.: Convergence test for SL, smooth profile, C = 0.25.

Rel. L2 Rel. L∞ Dissipation Dispersion Minerror error error error

h=0.1 3.47 e− 01 3.99 e− 01 2.48 e− 02 6.91 e− 02 −2.74 e− 01h=0.05 1.15 e− 01 1.41 e− 01 2.22 e− 03 8.55 e− 03 −1.56 e− 01

Table 3.7.: Convergence test for RKDG without slope limiting, smooth profile, C = 0.25.

These results show that at low Courant number the SLDG method does not suffer from the error amplificationthat is typical of SL methods (see, e.g., the analysis in [FF98]), while its accuracy is comparable to that of theRKDG method.

It is now important to evaluate the effects of the limiting procedure on the computed solution. With this aim,we compare the performance of the M version of SLDG method and RKDG method. For the latter, the slopelimiting monotonization described in [CHS90] was employed. Results are summarized in Tab. 3.8 for thesmooth advected profile and in Tab. 3.9 for the discontinous advected profile, respectively. Both cases werecomputed at resolution h = 0.1. It can be noticed that the SLDG method with FCT monotonization is farless diffusive than the slope limiting procedure used in the RKDG formulation. This can be seen also in theplots of the solutions displayed in Fig. 3.5. It can also be observed that the SLDG solution does not display theexcessive sharpening of smooth profiles reported e.g. in [SS96] in the case of more traditional applications ofthe FCT technique.

38



SLDG 4.27 e− 01 5.51 e− 01 6.25 e− 02 7.98 e− 02 −1.04 e− 15RKDG 8.05 e− 01 8.74 e− 01 3.17 e− 01 1.89 e− 01 4.44 e− 16

min(ch) max(ch) min(ch,0) max(ch,0)SLDG −1.31 e− 01 10− 5.39 e+ 00 −2.83 e− 19 10− 5.73 e+ 00RKDG −1.29 e− 02 10− 8.64 e+ 00 2.51 e− 12 10− 9.05 e+ 00

Table 3.8.: Comparison of SLDG and RKDG with monotonization, smooth profile, C = 0.25.


SLDG 4.48 e− 01 6.03 e− 01 2.18 e− 01 2.61 e− 01 −4.74 e− 16RKDG 8.04 e− 01 8.27 e− 01 9.73 e− 01 5.75 e− 01 −1.11 e− 15

min(ch) max(ch) min(ch,0) max(ch,0)SLDG −1.35 e− 01 10− 2.50 e+ 00 −7.33 e− 19 10− 3.09 e+ 00RKDG −2.13 e− 02 10− 7.62 e+ 00 5.75 e− 12 10− 8.34 e+ 00

Table 3.9.: Comparison of SLDG and RKDG with monotonization, discontinuous profile, C = 0.25.

The performance of the SLDG formulation at C = 3 is also analyzed. In this case, numerical quadraturesare performed by setting Le = 8, Lf = 6 and M = 10 in (3.6). Results are summarized in Tab. 3.10 forthe smooth advected profile without monotonization, computed at resolution h = 0.05. The SLDG results arecompared to those of a standard SL method with continuous P2 reconstruction. As is well known, the SLsolution is quite sensitive to the trajectory approximation technique. Thus, we report results in two extremecases of simple Euler approximation with substepping, indicated as SL(a), and of semi–Lagrangian advectioncomputed using the analytical trajectory, indicated as SL(b). In contrast, for the SLDG method only the simpleEuler approximation is used. It can be seen that the SLDG results are much less sensitive to the trajectoryapproximation method and that the SLDG errors are comparable to those of SL(b), while they are superior toSL(a). This greater accuracy of SLDG, however, corresponds to a higher computational cost, due to the factthat the numerical solution does not only involve reconstruction at the foot of the characteristic lines, but alongthese as well.


SLDG 7.31 e− 02 7.20 e− 02 2.55 e− 04 4.13 e− 03 −1.31 e− 15SL(a) 2.45 e− 01 2.44 e− 01 6.61 e− 03 4.37 e− 02 6.93 e− 02SL(b) 7.75 e− 02 7.40 e− 02 5.51 e− 04 4.47 e− 03 −4.20 e− 03

min(ch) max(ch) min(ch,0) max(ch,0)SLDG −7.68 e− 02 10− 5.33 e− 01 −4.85 e− 02 10− 6.70 e− 01SL(a) −2.49 e− 01 10− 8.80 e− 01 −2.36 e− 01 10− 9.74 e− 01SL(b) −2.37 e− 01 10− 7.59 e− 01 −2.34 e− 01 10− 8.12 e− 01

Table 3.10.: Comparison of SLDG with P1 reconstruction and SL with continuous P2 reconstruction, smoothprofile, C = 3.

39


Figure 3.5.: Monotonized solutions in the solid body rotation test case at C = 0.25, smooth profile (left) anddiscontinuous profile (right). First row: SLDG solution computed with P1 elements; second row:RKDG solution computed with P1 elements; third row: analytic solution.

40


3.5.3. Two–Dimensional Advection: Deformational Flow Tests

Two deformational flow tests are considered. The first one is the non divergent vortical velocity field introducedin [Dos84], and used by many authors to assess the accuracy of advection schemes, see, e.g., [NCS99, NM02].The second one is the well known test proposed by P. Smolarkiewicz in [Smo82].

For the Doswell test problem, a circular domain of radius R = 3 is considered, with a triangulation Th con-sisting of 2352 elements with Ndofs = 7056. In this case, numerical quadratures are performed by settingLe = 12, Lf = 6 and M = 18 in (3.6). The initial datum is a function taking two different constant values onthe upper and lower half of the computational domain, respectively, with a sharp transition zone in the middle,and the final time level is T = 20. The zero order degrees of freedom representing cell averages are displayed inFig. 3.6, as computed by the monotonized SLDG and RKDG schemes, at C = 2 and at C = 0.3, respectively.It can be observed that the monotonization approach proposed for SLDG leads to a much sharper interfaceand to much greater detail in the vortex roll-up zone, which is consistent with the error statistics shown inTab. 3.11. Together with the results in the previous section, these test cases lead to the conclusion that the FCTmonotonized SLDG method is superior to the standard RKDG method. Furthermore, beyond its application inconjunction to SLDG, the proposed FCT based monotonization technique might also be a useful improvementof monotonization techniques in the framework of generic DG approximations.

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Figure 3.6.: Doswell deformation flow test case: monotone SLDG solution computed with P1 elements atC = 2 (left) and limited RKDG reference solution computed with P1 elements at C = 0.3 (right).


SLDG 2.26 e− 01 1.49 e+ 00 1.02 e− 03 4.30 e− 02 −2.75 e− 15RKDG 3.61 e− 01 1.59 e+ 00 1.06 e− 02 1.01 e− 01 1.30 e− 15

min(ch) max(ch) min(ch,0) max(ch,0)SLDG −1− 4.76 e− 01 1 + 5.41 e− 01 −1− 1.33 e− 10 1 + 1.00 e− 13RKDG −1− 7.09 e− 02 1 + 9.74 e− 02 −1− 2.22 e− 16 1 + 2.22 e− 16

Table 3.11.: Errors for the SLDG solution (C = 2) and RKDG solution (C = 0.3) in the Doswell deformationflow test case.

For the Smolarkiewicz test problem, we use the SLDG scheme in its NM version on a structured triangulationwhich fits the boundaries of the convective cells of the flow, as done in the cited reference. Moreover, the

41


orientation of the mesh triangles is chosen in such a way to preserve the symmetry of the problem with respectto the axis x = 50. The number of elements is Nel = 3200, corresponding to Ndofs = 9600, and the time stepis chosen in order to have C = 4 approximately. In this case, numerical quadratures are performed by settingLe = 10, Lf = 6 and M = 24 in (3.6).

As pointed out in [SCP86], where an analytic solution for this test has been described, there are two flowregimes for which different evaluation criteria are appropriate. On a time scale of the order of the characteristicperiod of the flow, accurate numerical methods are assumed to reproduce the analytic solution correctly. Onthe other hand, on a much longer time scale, it can only be expected that the average behaviour of the analyticsolution is recovered. Fig. 3.7 (left) illustrates the computed solution at time level t = 37s, corresponding to 3/4of the characteristic period of the flow, thus in the first regime. Fig. 3.7 (right) shows the computed solution attime level t = 2500s, corresponding to 50 times the characteristic period of the flow, thus in the second regime.The results in the first regime compare well with the plots of the analytic solution presented in [SCP86]. Onthis time scale, results obtained with the monotonic version of the SLDG method are qualitatively very similarto those shown in Fig. 3.7. On the longer time scale, the results of the non–monotonic SLDG scheme arequalitatively similar to large scale average of the analytic solution. The monotonic SLDG scheme yields results(not displayed here) analogous to those of the FCT monotonized scheme discussed in [Smo82].

0 20 40 60 80 100−2

0

2

4

6

8

10

12

0 20 40 60 80 100

0

0.5

1

1.5

2

2.5

3

Figure 3.7.: Smolarkiewicz deformation flow test case. SLDG solution computed with P1 elements at Courantnumber 4. Left: time level t = 37s, first flow regime; right: time level t = 2500s, second flowregime. Upper row: three–dimensional plots of the computed solution. Bottom row: sections ofthe computed solution along the middle line of the computational domain.

42


Rel. L2 Rel. L∞ Dissipation Dispersionerror error error error

1D test 1.68 e− 02 2.43 e− 02 2.52 e− 08 1.44 e− 072D test 7.98 e− 02 1.44 e− 01 8.68 e− 05 1.89 e− 04

Table 3.12.: Advection diffusion tests: results obtained with SLDG in one- and two-dimensional cases.

3.5.4. Tests on the Advection–Diffusion Equation

In this section, we address the issue of including a discretization of the diffusion term into the SLDG context.Given the structure outlined in the previous sections, the Discontinuous Galerkin methods developed for ellipticproblems discussed in Sect. 2.3 appear to be a natural choice. They use the same discontinuous piecewisepolynomial representation of the unknown concentration considered for the SLDG method, they guaranteemass conservation at the element level and can reach high order accuracy. In a preliminary implementation,we choose the Interior Penalty (IP) formulation discussed in [ABCM02] for the stationary case and appliedin [LS03] to time dependent problems. A simple forward Euler method is considered for the time stepping.This choice results in stability and accuracy restrictions for the overall scheme, whose maximum available timestep is dictated by the diffusion process. Other time stepping techniques for the diffusion term could also beconsidered. Furthermore, it is to be remarked that, for simplicity, diffusive terms are included here by means ofan operator splitting approach. We do not consider this an optimal solution and more appropriate ways to dealwith the diffusion term in SLDG are currently being investigated. On the other hand, operator splitting is oftenused to include diffusion in many existing models, so that the present results are an indication of what can beexpected if the SLDG formulation were applied within one of these models.

Two numerical experiments are carried out to validate the proposed formulation for advection dominated flowregimes. In both cases, a constant viscosity ν is considered. The first test considers one–dimensional constantvelocity advection of an initial Gaussian profile with zero mean and standard deviation σ0 = 1 in presenceof diffusion. The flow field, the diffusion coefficient and the final time level are chosen in such a way thatthe final profile has mean and standard deviation equal to 53σ0 and 3σ0, respectively. The Peclet number isUL/2ν = 343, where U denotes the magnitude of the advective velocity and L denotes a typical length scale.The Courant number is C = 1.5, while ν∆t/h2 = 0.16. The stabilization parameter for the IP scheme is set to1 and P1 elements are considered. Results for the M version of SLDG advection are shown in Fig. 3.8. For thetwo–dimensional case, the solid body rotation of an initial Gaussian profile is considered with mean (0.5, 1)and standard deviation σ0 = 0.0427, under the same setting as in Sect. 3.5.2. The grid size is such that theCourant number is approximately equal to C = 3. Five rotations are performed and the diffusion coefficientis chosen in such a way that the Peclet number is UL/2ν = 3900 approximately while ν∆t/A = 0.02approximately, where A denotes the average element area. The solution at final time has standard deviationequal to σ0 = 0.077. The results for the M version of SLDG advection are shown in Fig. 3.9. The error normsfor both cases are displayed in Tab. 3.12. We can observe that the errors obtained are approximately of the samemagnitude as in purely advective tests carried out at analogous space-time resolutions (see, e.g., Tab. 3.10).Mass is conserved up to machine precision and no significant difference is observed when considering theNM version of the scheme. Furthermore, in the two–dimensional test the error in the standard deviation ofthe computed solution can be estimated at approximately 10% of the exact value. Thus, even consideringthe limitations of the approach used to introduce diffusion, the resulting method for the advection-diffusionequation appears to be sufficiently accurate for many practical applications.

43


−10 −5 0 5 100

0.1

0.2

0.3

0.4

0.5

35 40 45 50 55 60 65 700

0.05

0.1

0.15

Figure 3.8.: One–dimensional advection diffusion test cases. Left: initial datum. Right: computed P1 solution(solid line, mean values represented by circles) and exact solution (dashed line).

Figure 3.9.: Two–dimensional advection diffusion test cases. Left: exact solution. Right: computed P1 solu-tion.

3.6. Open Issues and Further Developments

One major open issue with respect to the SLDG method appears to be the characterization of the accuracyrequired in the space-time quadrature formulae introduced in Sect. 3.2.4. A precise evaluation of the efficiencyof the SLDG method compared to other more mature methods can be feasible only once the above issue isproperly addressed. The accuracy required in the quadrature formulae for the approximate computation of thefluxes and volume integrals in equation (3.2) appears to be related to the local Courant number. Precisely,for low Courant numbers, a smaller number of quadrature points appears to be sufficient to achieve the samelevel of accuracy, compared to the numbers necessary for higher Courant number cases. One possibility toimprove computational efficiency could be to choose the number of quadrature points locally in space and timeas a function of the Courant number. Finally, a major theoretical development that also appears feasible is toprovide a convergence and stability proof for the SLDG method along the lines of [FF98], which should be ableto give a more rigorous basis to the empirical finding on the better convergence rate of the SLDG formulationcompared to the classical SL approach discussed in Sect. 3.5.

44

Chapter 4.

The Semi–Implicit Discontinuous GalerkinMethod

In this chapter the application of the DG method to the approximation of the Navier–Stokes Equations for atmospheric flows is considered, with particular emphasis on theextension of the semi–implicit time discretization to the DG framework. In Sect. 4.2 theNavier–Stokes Equations are reformulated in terms of deviations from a reference state toavoid problems associated to roundoff errors in the numerical approximation. Section 4.3describes the treatment of the open boundary conditions, which is a critical element inthe applications. The SI time discretization is illustrated in Sect. 4.4, while the DG high–order space discretization is described in details in Sect. 4.5. Section 4.6 summarizes thefully discrete problem. Section 4.7 provides some details concerning the implementationof the method and also summarizes the filtering procedure introduced to ensure stabilityof the scheme. Finally, section 4.8 illustrates how it is possible to reformulate the linearsystem associated with the SI time discretization in such a way to end up with a problemof elliptic type in the sole pressure unknown, which significantly increases the efficiencyof the scheme.

4.1. Introduction

In this chapter, we consider the application of the DG method to the approximation of the Navier–Stokes equa-tions for atmospheric flows. In particular, we propose a method to combine the SI time discretization, which iswidely employed in the context of finite difference formulations, with a DG–based spatial discretization. Ouraim is to exploit the capability of the SI time discretization in improving the efficiency of atmospheric models(see Sect. 1.2) and the high–order accuracy and conservation properties provided by the DG finite elementapproximation (see Chapt. 2). Concerning with the spatial discretization, we follow the framework proposedin [GHW02] and [GW], where particular emphasis is given to the high–order representation of the unknowns.In particular, this requires a different definition of the finite dimensional space Vh with respect to the onegiven in Chapt. 3. Concerning with the SI time discretization, we follow the general framework described inSect. 1.2.2 to devise a method which is unconditionally stable with respect to acoustic and gravity waves. Toour knowledge, the resulting semi–Implicit DG formulation represents a novel contribution, which, besidesbeing suited for atmospheric flow problems, may be useful also in the context of smaller scale, low Machnumber fluid dynamics simulations. We notice here that the scheme proposed in this thesis is different from

Chapter 4. The Semi–Implicit Discontinuous Galerkin Method

the one proposed in [DF04] despite the fact that both of them are referred to as “semi–Implicit DiscontinuousGalerkin” formulations. The method proposed by Dolejsı and Feistauer relies in fact on a careful handling ofthe nonlinear problem arising from a fully implicit time stepping, and a key ingredient of the method is thechoice of a particular numerical flux. This approach is thus suitable for high Mach number flows. On thecontrary, in our approach only the terms responsible for acoustic and gravity waves are treated implicitly, andthere is much freedom in the choice of the numerical flux. As a result, the scheme we propose is effective forthe treatment of low Mach number flows.

The detailed outline of this chapter is as follows. In Sect. 4.2 the Navier–Stokes Equations are reformulated interms of deviations from a reference state to avoid problems associated with roundoff errors in the numericalapproximation, in a way which is classical in atmospheric modeling. Section 4.3 describes the treatment of theopen boundary conditions, which is a critical element in the applications at hand. The SI time discretizationis then illustrated in Sect. 4.4, while the DG high order space discretization is described in details in Sect. 4.5.In particular, we introduce in Sect. 4.5.1 the approximate quadrature rules playing a central role both in theimplementation of the scheme and in the static condensation procedure described later. Section 4.6 summarizesthe fully discrete problem. Section 4.7 provides some details concerning the implementation of the method,exploiting the structure of the finite element space. In particular, in Sect. 4.7.1 it is shown how the schemecan be interpreted as a spectral element formulation of collocation type, while the filtering procedure which isnecessary to ensure stability of the method is described in Sect. 4.7.3. The concluding section 4.8 illustrateshow it is possible to reformulate the linear system associated with the SI time discretization in such a way toend up with a problem of elliptic type in the sole pressure unknown, which significantly increases the efficiencyof the scheme.

4.2. The Navier–Stokes Equations for Stratified Flows

The NS equations have been introduced in Eq. (2.20). It is now convenient to expand (2.20)1 componentwise.Thus, considering a two–dimensional problem, we let

V =[UW

], Fv =

0 0

F vU,x F v

U,z

F vW,x F v

W,z

F vE,x F v

E,z

(4.1)

and obtain

∂ρ

∂t= −∂U

∂x− ∂W

∂z

∂U

∂t= − ∂

∂x

[U2

ρ

]− ∂

∂z

[UW

ρ

]− ∂p

∂x+∂F v

U,x

∂x+∂F v

U,z

∂z

∂w

∂t= − ∂

∂x

[UW

ρ

]− ∂

∂z

[W 2

ρ

]− ∂p

∂z+∂F v

W,x

∂x+∂F v

W,z

∂z− ρ g

∂E

∂t= − ∂

∂x

[UH

ρ

]− ∂

∂z

[WH

ρ

]+∂F v

E,x

∂x+∂F v

E,z

∂z

(4.2)

where p = p(ρ, U,W,E) through the equation of state. Notice that, although for practical applications aturbulence closure relation has to be considered to define the viscosity coefficients, in the present work weassume that these latter are known constants. When solving system (4.2) for atmospheric flows, it can be

expected that the flow is nearly hydrostatic, i.e. in the vertical momentum equation the two terms∂p

∂zand ρg

are much larger than the remaining ones. This can cause instabilities in the numerical approximation of theproblem, due to cancellation of significant digits. To avoid this effect, problem (4.2) can be reformulated in

46

4.3. Treatment of the Open Boundary Conditions

terms of deviations from a constant reference state. Thus, we introduce ρ, U , W , E and p = p(ρ, U,W,E)such that:

U = 0; W = 0;

∂ρ

∂x= 0;

∂E

∂x= 0;

dp

dz= −ρ g.

(4.3)

Notice that relations (4.3)2 immediately imply that∂p

∂x= 0. Upon defining ρ′ = ρ − ρ, E′ = E − E and

p′ = p− p, we obtain

∂ρ′

∂t= −∂U

∂x− ∂W

∂z

∂U

∂t= − ∂

∂x

[U2

ρ

]− ∂

∂z

[UW

ρ

]− ∂p′

∂x+∂F v

U,x

∂x+∂F v

U,z

∂z

∂W

∂t= − ∂

∂x

[UW

ρ

]− ∂

∂z

[W 2

ρ

]− ∂p′

∂z+∂F v

W,x

∂x+∂F v

W,z

∂z− ρ′ g

∂E′

∂t= − ∂

∂x

[UH

ρ

]− ∂

∂z

[WH

ρ

]+∂F v

E,x

∂x+∂F v

E,z

∂z.

(4.4)

Notice that the equation of state allows for an expression of p′ which is independent from ρ, E. In fact, wehave

p′ =R

cv

(E′ − 1

2V 2

ρ− ρ′gz

), (4.5)

and no cancellation problems occur in evaluating the pressure perturbation. Problem (4.4) has the advantagethat all the terms in the vertical momentum equations are of the same order of magnitude. For this reason, (4.4)will be used instead of (4.2) in the remainder of this chapter.

4.3. Treatment of the Open Boundary Conditions

As discussed in Sect. 1.2.3, in practical applications, it is usually necessary to truncate the computational do-main with artificial boundaries, not corresponding to any physical entity. Ideally, an “open boundary” conditionis desired on these boundaries, avoiding any reflection of outgoing signals. A simple and robust solution is rep-resented by an absorbing layer, also known as sponge layer. The underlying idea is illustrated in Fig. 4.1,which shows the damping of an outgoing wave, while minimal reflection takes place partly within the layeritself and partly on the domain boundary. The definition of the structure of the absorbing layer is a criticalparameter for the performance of the model, and a compromise is necessary between conflicting requirements:thick sponge layers are more effective in preventing reflection, but result as well in a waste of computing re-sources, since the numerical solution does not have any physical meaning within the layer. Also, large dampingcoefficients can result in reflection on the absorbing layer itself, while too small damping coefficients can notprevent reflection at the domain boundaries. In [KL78] a method is described to optimize the structure of theabsorbing layer when the primitive variable formulation (1.15) is considered. To apply these results to theconservative formulation (4.4), we can in principle perform a change of variables from conservative to prim-itive variables, apply the damping coefficients τπ and τ as in (1.37) and then transform back to conservativevariables. In practice, this can be done as follows: denoting by qb = [ρb, Ub, Wb, Eb]T a known boundary

47


Figure 4.1.: Schematic representation of the absorbing layer: an outgoing wave (red) is first damped while trav-eling in the layer, which also yields some reflection (blue), then reflected at the domain boundary.This reflected wave (green) undergoes further damping while traveling toward the inner domain.

datum, we modify system (4.4) to obtain

∂ρ′

∂t= −∂U

∂x− ∂W

∂z− τρ · (q− qb)

∂U

∂t= − ∂

∂x

[U2

ρ

]− ∂

∂z

[UW

ρ

]− ∂p′

∂x+∂F v

U,x

∂x+∂F v

U,z

∂z− τU · (q− qb)

∂W

∂t= − ∂

∂x

[UW

ρ

]− ∂

∂z

[W 2

ρ

]− ∂p′

∂z+∂F v

W,x

∂x+∂F v

W,z

∂z− ρ′ g − τW · (q− qb)

∂E′

∂t= − ∂

∂x

[UH

ρ

]− ∂

∂z

[WH

ρ

]+∂F v

E,x

∂x+∂F v

E,z

∂z− τE · (q− qb),

(4.6)

where the vectors τρ, τU , τW and τE are yet to be defined. With this purpose, we now linearize the transfor-mation from conservative to primitive variables in a neighborhood of the boundary state qb, obtaining

π′

uwθ′

'

γ−1γ

1p00

(p00pb

) 1γ

0 0 0

0 1 0 00 0 1 0

−γ−1γ

(p00pb

) γ−1γ Tb

pb0 0

(p00pb

) γ−1γ

·

Rcv

(12

V 2b

ρ2b− g z

)− R

cv

Ub

ρb− R

cv

Wb

ρb

Rcv

−Ub

ρ2b

1ρb

0 0

−Wb

ρ2b

0 1ρb

0

1cvρ2

b

(V 2

b

ρb− Eb

)− 1

cv

Ub

ρ2b

− 1cv

Wb

ρ2b

1cvρb

ρ′

UWE′

.

Then, denoting by M the resulting transformation matrix in the above equation, we compute the followingmatrix

T = M−1

τπ 0 0 00 τ 0 00 0 τ 00 0 0 τ

M, (4.7)

and identify the vectors τρ, τU , τW and τE as the rows of T .

48

4.4. Semi–Implicit Splitting of the Navier–Stokes Equations in Conservation Form

Once the absorbing layer has been introduced, the particular boundary condition on ∂Ω has in practice no effecton the computed solution, so that, because of the ease of implementation, we impose the Dirichlet boundarycondition q = qb on ∂Ω.

For completeness, we notice that, although introducing an absorbing layer is a common solution to handle openboundary conditions, more sophisticated alternatives are possible. A method based of the Fourier transformof the numerical solution is discussed in [KD83]. Also it is possible to resort to the Perfectly Matched Layerapproach. This has been done in [GN03] for the Shallow Water Equations, and the application to the NSEquations is presently under investigation.

4.4. Semi–Implicit Splitting of the Navier–Stokes Equations inConservation Form

In this section, we consider the SI time semi–discretization of (4.6). With this purpose we need to introducesuitable operators L and L 0 as well as constant terms f and f0 as in Sect. 1.2.2. Following [KSD00], we thusdefine

L q =

−∂U∂x

− ∂W

∂z

− ∂

∂x

[R

cv(E′ − ρ′gz)

]− ∂

∂z

[R

cv(E′ − ρ′gz)

]− ρ′g

− ∂

∂x

[hU]− ∂

∂z

[hW

]

f =

0

0

0

0

(4.8)

where h = 1ρ

(E + p

), and

L 0q = −T q, f0 = T qb. (4.9)

A theoretical characterization of the stability properties of the time discretization based on the operators in (4.8)will now be presented for the case of a simplified problem, following [Dur99]. The stability of the scheme fornonlinear problems, including the effect of the absorbing boundary conditions represented by (4.9), will bedemonstrated experimentally in Chapt. 5.

4.4.1. Stability Analysis in One Spatial Dimension

For simplicity, in this section all viscous effects will be neglected, so that the system of the Euler Equationswill be considered. In addition, when multistep time integration schemes as those in Tab. 1.1 are employedto integrate the EE, the stability analysis tends to become rather involved even in the linear case, especiallywhen taking into account the atmospheric stratification. Hence, we analyze here in the Von Neumann sense thestability of (4.8) for the time discretization of two problems which, despite being simpler than the complete setof the Euler Equations, still reproduce some crucial characteristics of the original system, namely, the presenceof acoustic waves, in the first case, and the simultaneous presence of fast moving waves and advection, in thesecond case. These two problems thus provide necessary conditions for stability in the general case.

The leapfrog scheme is adopted, with θ = 12 . In the first case, we consider the linearized EE in one spatial

49


dimension and assume no stratification and no mean flow, i.e. we consider the problem

∂ρ′

∂t= −∂U

∂x

∂U

∂t= −R

cv

∂E′

∂x

∂E′

∂t= −h∂U

∂x

(4.10)

with h = const. Dropping the primes, the SI time discretization of problem (4.10) reads

ρn+1 − ρn−1

2∆t= −1

2

(∂Un+1

∂x+∂Un−1

∂x

)Un+1 − Un−1

2∆t= −1

2R

cv

(∂En+1

∂x+∂En−1

∂x

)En+1 − En−1

2∆t= −1

2h

(∂Un+1

∂x+∂Un−1

∂x

).

(4.11)

Letting now (ρ, U,E)(x, t) = (ρ, U , E)ei(kx−ωt), we obtain the dispersion relation

ω1 = 0; ω2,3 = ± 1∆t

atan (a∆tk) . (4.12)

In order to have non amplifying solutions, we need ω ∈ R, which, from (4.12), is always the case. The schemeis thus stable.

In the second case, we consider the system of the linearized Shallow Water Equations in one spatial dimension.This system models the motion of a shallow fluid layer in presence of gravity. Besides representing a relevantproblem in environmental fluid dynamics on their own (see for instance [GHW02]), the Shallow Water Equa-tions are closely related to the Euler Equations in that they include advective terms and support fast movinggravity waves. Denoting by η the free surface height, by u the horizontal velocity and setting U = ηu, uponintroducing a constant reference state η, u and letting as usual η = η + η′ and U = ηu + U ′, the linearizedShallw Water Equations read

∂η′

∂t= −∂U

′

∂x

∂U ′

∂t= −2u

∂U ′

∂x− u2 ∂η

′

∂x+ η

∂η′

∂x.

(4.13)

Dropping the primes for simplicity, application of the SI time discretization to (4.13) yields the followingsemidiscretized problem:

ηn+1 − ηn−1

2∆t= −1

2

(∂Un+1

∂x+∂Un−1

∂x

)Un+1 − Un−1

2∆t= −2u

∂Un

∂x− u2 ∂η

n

∂x+

12η

(∂ηn+1

∂x+∂ηn−1

∂x

).

(4.14)

Letting now (η, U)(x, t) = (η, U)ei(kx−ωt), we obtain the dispersion relation

(ω − uk) = k2(ηC2 − u2C + u2

), (4.15)

where ω = 1∆t sinω∆t and C = cos(ω∆t). After some algebraic manipulations, it can be seen that a sufficient

condition for ω ∈ R is(1 +

√2)|u| ≤

√η,

this implying that the solution is non amplifying irrespective of the amplitude of the time step ∆t.

50

4.5. Spatial Discretization


In this section, we complete the discretization of problem (4.6) by introducing a high–order, nodal, DG spa-tial discretization for the operator S (q), defined by the right–hand side of (4.6), and the operators L q andL 0q defined in (4.8) and (4.9), respectively. This will be accomplished by following the general frameworkpresented in Chapt. 2 and choosing a particular set of basis functions and a numerical quadrature rule. Thissection closely follows [GHW02].

4.5.1. High–Order Polynomial Spaces

Let Th denote a grid of nonoverlapping curvilinear quadrilateral elementsK which are images of the referenceelement K = [−1, 1]2 under smooth, bijective maps FK

∀K ∈ Th : K = FK(K).

We denote by ∂K and nbe,∂ bK the boundary of K and the outward unit normal vector on each edge e ∈ ∂K,respectively. The notation x = FK(ξ), with x = (x, z), ξ = (ξ, ζ) will be used. We also associate with eachlocal map the transformation Jacobian JK = dx

dξ

∣∣∣K

and the determinant |JK |. Although not essential, we willassume that Th is conforming, that is, given K1, K2 ∈ Th we either have that K1 ∩ K2 is empty, or it is avertex or a complete edge e ∈ Eh (see Fig. 4.2).

FK

K

x

zξ

ζ

K

Th

Figure 4.2.: Each element K ∈ Th (green), in the physical plane x, z, is the image under the map FK of thereference element K = [−1 , 1]2 in the reference plane ξ, ζ.

Following a standard procedure in the context of finite element formulations, we will first introduce a set ofpolynomial shape functions on the reference element and than compose them with the local maps to obtainthe basis functions of the finite dimensional space Vh. Concerning the choice of the shape functions, thediscontinuous nature of Vh allows maximum freedom. Basically, two main alternatives are possible: modalbasis functions and nodal basis functions. Modal basis functions provide a hierarchical representation, whereeach basis function is associated with a given wavelength. On the other hand, nodal basis functions allow foran immediate physical interpretation of the expansion coefficients as values of the unknown variable at a set ofgrid nodes. In this presentation, nodal basis functions will be adopted, with the exception of Sect. 4.7.3, wheretransformations between nodal and modal representations are required.

The logical square structure of the reference element K significantly simplifies the construction of high–orderpolynomial bases, since the multidimensional basis can be obtained as tensor–product of the one–dimensionalbasis. For a positive integer k, we introduce the following two bases for Pk([−1, 1]): ϕi(ξ)k

i=0 is the Lagran-gian (i.e. nodal) basis associated with an arbitrary set of nodes ξi ∈ [−1, 1], for i = 0, . . . , k, while ψi(ξ)k

i=0

51


is the Legendre (i.e. modal) basis (see [CHQZ06]). We also define

Φij(ξ) = ϕi(ξ)ϕj(ζ)

and

ΦK,ij(x) =

(Φij F−1

K )(x), x ∈ K

0, x /∈ K.

For simplicity, a global index I , ranging from 1 to N = (k + 1)2Nel, is also biunivocally associated with thelocal index (K, ij). We set Vh = span ΦI, I = 1, . . . ,N .

Although, in principle, several choices for the nodes ξi are possible, a convenient one is represented by theLegendre–Gauss–Lobatto (LGL) points, defined as the roots of the polynomial

(1− ξ2

)ψ′k(ξ). As a matter of

fact, these points are endowed with a Gaussian quadrature rule that can be exploited to improve the efficiencyof the resulting scheme (see [CHQZ06]). Concerning now the evaluation of the integrals introduced by the DGformulation, the following approximate quadrature rules will be adopted

∫K

φ(x)dx =∫

bK φ(x(ξ))|JK(ξ)|dξ ≈k∑

i,j=0

φ(x(ξi, ζj))|JK(ξi, ζj)|wiwj (4.16)

and ∫∂K

φ(σ)dσ =∑

be∈∂ bK∫

be φ(x(σ)) |JK(σ)| ||J−TK (σ)nbe,∂ bK ||2 dσ

≈k∑

i=0

φ(x(ξi, ζ0)) |JK(ξi, ζ0)| ||J−TK (ξi, ζ0)nbe1,∂ bK ||2 wi

+k∑

j=0

φ(x(ξk, ζj)) |JK(ξk, ζj)| ||J−TK (ξk, ζj)nbe2,∂ bK ||2 wj

+k∑

i=0

φ(x(ξi, ζk)) |JK(ξi, ζk)| ||J−TK (ξi, ζk)nbe3,∂ bK ||2 wi

+k∑

j=0

φ(x(ξ0, ζj)) |JK(ξ0, ζj)| ||J−TK (ξ0, ζj)nbe4,∂ bK ||2 wj

(4.17)

where φ is a generic function piecewise continuous on Th, || · ||2 is the Euclidean norm of a vector, ξi and ζiare Legendre–Gauss–Lobatto points previously introduced and wi are the associated weights, defined as

wi =2

k(k + 1)

(1

ψk(ξi)

).

In the following, for i, j = 0, . . . , k, we will let

wK,ij = |JK(ξi, ζj)|wiwj ,

ωeK,ij =

|JK(ξi, ζj)| ||J−T

K (ξi, ζj)nbe,∂ bK ||2 wi, (j, e) ∈ (0, e1) , (k, e3)

|JK(ξi, ζj)| ||J−TK (ξi, ζj)nbe,∂ bK ||2 wj , (i, e) ∈ (0, e2) , (k, e4)

0 otherwise.

Notice that the degree of exactness of the quadrature rules (4.16) and (4.17) is 2k − 1.

52


4.5.2. Discontinuous Galerkin Discretization

The strong form DG discretization of problem (4.6) reads:find qh(·, t) = (ρ′h, Uh,Wh, E

′h)(·, t) ∈ (Vh)4 such that for all t ∈ [0, T ] and ∀K ∈ Th, I = 1, . . .N ,

d

dt

∫K

ΦIρ′h dx = −

∫K

ΦI

(∂F e

ρ,x

∂x+∂F e

ρ,z

∂z

)dx

+∫

∂K

ΦI

((F e

ρ,x − F eρ,x

)nx +

(F e

ρ,z − F eρ,z

)nz

)dσ

−∫

K

ΦIτρ · (qh − qb) dx

d

dt

∫K

ΦIUhdx = −∫

K

ΦI

(∂F e

U,x

∂x+∂F e

U,z

∂z

)dx

+∫

K

(∂ΦI

∂xF v

U,x +∂ΦI

∂zF v

U,z

)dx

+∫

∂K

ΦI

((F e

U,x − F eU,x

)nx +

(F e

U,z − F eU,z

)nz

)dσ

−∫

∂K

ΦI

(F v

U,xnx + F vU,znz

)dσ

−∫

K

ΦIτU · (qh − qb) dx

d

dt

∫K

ΦIWh dx = −∫

K

ΦI

(∂F e

W,x

∂x+∂F e

W,z

∂z

)dx−

∫K

ΦIρ′hg dx

+∫

K

(∂ΦI

∂xF v

W,x +∂ΦI

∂zF v

W,z

)dx

+∫

∂K

ΦI

((F e

W,x − F eW,x

)nx +

(F e

W,z − F eW,z

)nz

)dσ

−∫

∂K

ΦI

(F v

W,xnx + F vW,znz

)dσ

−∫

K

ΦIτW · (qh − qb) dx

d

dt

∫K

ΦIE′h dx = −

∫K

ΦI

(∂F e

E,x

∂x+∂F e

E,z

∂z

)dx

+∫

K

(∂ΦI

∂xF v

E,x +∂ΦI

∂zF v

E,z

)dx

+∫

∂K

ΦI

((F e

E,x − F eE,x

)nx +

(F e

E,z − F eE,z

)nz

)dσ

−∫

∂K

ΦI

(F v

E,xnx + F vE,znz

)dσ

−∫

K

ΦIτE · (qh − qb) dx

(4.18)

whereF e

ρ,x = Uh F eρ,z = Wh

F eU,x =

U2h

ρh+ p′h F e

U,z =UhWh

ρh

F eW,x =

UhWh

ρhF e

W,z =W 2

h

ρh+ p′h

F eE,x =

UhHh

ρhF e

E,z =WhHh

ρh

53


and the numerical hyperbolic fluxes F e·,· are such that

he,K =

F e

ρ,x F eρ,z

F eU,x F e

U,z

F eW,x F e

W,z

F eE,x F e

E,z

· ne,∂K ,

he,K being the Rusanov numerical flux (2.22). A natural choice is represented by

[F e

ρ,x

F eρ,z

]= Vh+

|λ|2

[[ρ′h]]

[F e

U,x

F eU,z

]=Uh

ρhVh

+[p′h

0

]+|λ|2

[[Uh]]

[F e

W,x

F eW,z

]=Wh

ρhVh

+[

0p′h

]+|λ|2

[[Wh]]

[F e

E,x

F eE,z

]=Hh

ρhVh

+|λ|2

[[E′h]]

(4.19)and |λ| is defined in (2.22). Concerning the viscous terms, the Bassi and Rebay method, which is a particularcase of the LDG approach described in Sect. 2.3 and 2.4, is adopted. This requires the introduction of anauxiliary variable Sh, defined at each time level by (2.24)2, and of the numerical fluxes (2.23). The termsF v·,· and F v

·,· are then obtained as the components of the viscous flux Fv(qh,Sh) and viscous numerical fluxFv(q, S), respectively, as in (4.1). Equation (4.18) defines the discrete operator Sh (qh).

4.5.3. The Linear Problem

Proceeding as done in Sect. 4.4 to recover (4.8) and (4.9) from the continuous problem (4.6), we identifywithin (4.18) the following linear operators:

(Lhqh)I =

−∫

K

ΦI

(∂Uh

∂x+∂Wh

∂z

)dx

+∫

∂K

ΦI

((FL

ρ,x − FLρ,x

)nx +

(FL

ρ,z − FLρ,z

)nz

)dσ

−Rcv

∫K

ΦI∂

∂x[E′

h − ρ′hgz] dx

+∫

∂K

ΦI

((FL

U,x − FLU,x

)nx +

(FL

U,z − FLU,z

)nz

)dσ

−Rcv

∫K

ΦI∂

∂z[E′

h − ρ′hgz] dx−∫

K

ΦIρ′hg dx

+∫

∂K

ΦI

((FL

W,x − FLW,x

)nx +

(FL

W,z − FLW,z

)nz

)dσ

−∫

K

ΦI

(hh

(∂Uh

∂x+∂Wh

∂z

)+dhh

dzWh

)dx

+∫

∂K

ΦI

((FL

E,x − FLE,x

)nx +

(FL

E,z − FLE,z

)nz

)dσ

(4.20)

54

4.6. The Fully Discrete Problem

and

(L 0

h qh

)I

= −

∫K

ΦIτρ · qh dx

∫K

ΦIτU · qh dx

∫K

ΦIτW · qh dx

∫K

ΦIτE · qh dx

, (4.21)

for I = 1, . . . ,N . In (4.20) h = Hρ , the linear fluxes are defined as follows:

FLρ,x = Uh FL

ρ,z = Wh

FLU,x = p′h FL

U,z = 0

FLW,x = 0 FL

W,z = p′h

FLE,x = hUh FL

E,z = hWh

and the linear numerical fluxes are[FL

ρ,x

FLρ,z

]= Vh+

|λ|2

[[ρ′h]]

[FL

U,x

FLU,z

]=[p′h

0

]+|λ|2

[[Uh]]

[FL

W,x

FLW,z

]=[

0p′h

]+|λ|2

[[Wh]]

[FL

E,x

FLE,z

]= h Vh+

|λ|2

[[E′h]]

where |λ| =√γRT .

Remark 4.5.1 We notice that the linear operators Lhqh and L 0h qh represent the approximate counterparts

using the DG finite element method supplied with the Rusanov numerical flux of the corresponding space–continuous operators L q and L 0q introduced in (4.8) and (4.9).

4.6. The Fully Discrete Problem

The fully discrete space–time approximation of problem (4.6) can be obtained by properly substituting into themultistep time advancing algorithm illustrated in Sect. 1.2.2 the discrete operators Sh(qh), Lhqh and L 0

h qh

introduced in (4.18), (4.20) and (4.21), respectively.

4.7. Implementation Details

In this section, we address the issue of the approximate computation of the several integral terms which appearin the fully discrete problem described in the previous sections. With this purpose, we employ here the quadra-ture rules (4.16) and (4.17) by taking advantage of the fact that the quadrature nodes coincide with the nodes

55


of the Lagrangian basis. This allows us to show in Sect. 4.7.1 that the strong form DG formulation can beregarded as a classical spectral finite element method of collocation type. In Sect. 4.7.2, we demonstrate howthe tensor product structure of the finite element basis allows for an efficient implementation of the resultingscheme. Finally, section 4.7.3 summarizes the filtering procedure introduced to ensure stability of the scheme.

4.7.1. Collocation Form of the Semi–Implicit Discontinuous–Galerkin Method

From the definition of the basis function ΦI and from (4.16) and (4.17), it immediately follows that, for ageneric function φ piecewise continuous on Th, we have∫

K

ΦI(x)φ(x) dx ≈ wIφ(xI)

and ∫∂K

ΦI(σ)φ(σ) dσ ≈∑

e∈∂K

ωeIφ(xI).

In the following, the notation EK,ij = e ∈ ∂K : xK,ij ∈ e will be used. Excluding the viscous terms,problem (4.18) yields the following ODE system for I = 1, . . . ,N :

dρ′Idt

= −(∂F e

ρ,x

∂x

)I

−(∂F e

ρ,z

∂z

)I

+∑e∈EI

ωeI

wI

[(F e

ρ,x − F eρ,x

)InxI

+(F e

ρ,z − F eρ,z

)InzI

]− τρ · (qI − qbI

)

dUI

dt= −

(∂F e

U,x

∂x

)I

−(∂F e

U,z

∂z

)I

+∑e∈EI

ωeI

wI

[(F e

U,x − F eU,x

)InxI

+(F e

U,z − F eU,z

)InzI

]− τU · (qI − qbI

)

dWI

dt= −

(∂F e

W,x

∂x

)I

−(∂F e

W,z

∂z

)I

+∑e∈EI

ωeI

wI

[(F e

W,x − F eW,x

)InxI

+(F e

W,z − F eW,z

)InzI

]− τW · (qI − qbI

)

dE′I

dt= −

(∂F e

E,x

∂x

)I

−(∂F e

E,z

∂z

)I

+∑e∈EI

ωeI

wI

[(F e

E,x − F eE,x

)InxI

+(F e

E,z − F eE,z

)InzI

]− τE · (qI − qbI

)

(4.22)

where (φ)I is a shorthand notation for φ(xI). Problem (4.22) represents a spectral finite element method ofcollocation type. Notice that, due to the discontinuous nature of the basis functions, an additional tendencyterm is associated with the interelement jumps.

Concerning the viscous flux terms, it is not computationally convenient to perform counterintegration by parts,so that they are not recast under a collocation–type form but are instead computed as

∫K

(∂ΦI

∂xF v

U,x +∂ΦI

∂zF v

U,z

)dx =

k∑i,j=0

(∂ΦI

∂xF v

U,x +∂ΦI

∂zF v

U,z

)K,ij

wK,ij (4.23)

with analogous expressions for the remaining equations.

56


Concerning the auxiliary variable Sh, Eq. (2.24)2 yields

Sρ,xI=(∂ρh

∂x

)I

−∑e∈EI

ωeI

wI(ρI − ρI)nxI

Sρ,zI=(∂ρh

∂z

)I

−∑e∈EI

ωeI

wI(ρI − ρI)nzI

(4.24)

with analogous relations for the remaining components S·,·I of SI . Since ρ = ρ, we can rewrite (4.24) as[Sρ,xI

Sρ,zI

]= (∇ρh)I −

∑e∈EI

12ωe

I

wI[[ρh]]I . (4.25)

Remark 4.7.1 In [BR97a] it is pointed out that Sh is the sum of an interface contribution and a volumecontribution. Relation (4.25) can be regarded as the collocation–type counterpart of Eq. (18) in the citedreference. The same consideration applies to (4.25) with regard to Eq. (3.17) in [ABCM02], where the liftingoperator r is given by

r ([[ρh]]) =N∑

I=1

(−∑e∈EI

12ωe

I

wI[[ρh]]I

)ΦI .

Notice also that ωeI and wI are proportional respectively to h and h2, so that ωe

I

wIis proportional to h−1 and

thus the quantity ωeI

wI[[ρh]] is dimensionally consistent with ∇ρh.

4.7.2. Efficient Evaluation of The Integrals

From the previous section, it is clear that the implementation of the semi–Implicit DG method requires thecomputation of the derivatives of the hyperbolic fluxes, of the unknown qh and of the basis functions ΨI at thequadrature nodes. The derivatives of the hyperbolic fluxes can be expressed in terms of the derivatives of theunknowns using the chain rule. For instance, we have(

∂F eρ,x

∂x

)I

=(∂Uh

∂x

)I

,(∂F e

U,x

∂x

)I

= 2UI

ρI

(∂Uh

∂x

)I

− U2I

ρ2I

(∂ρh

∂x

)I

+(∂p′h∂x

)I

,

and so forth. The derivatives in the physical space∂

∂xand

∂

∂zcan be computed from the derivatives in the

reference space as∂

∂x=∂ξ

∂x

∂

∂ξ+∂ζ

∂x

∂

∂ζ,

so that the building blocks for the implementation are the derivatives of the unknowns and of the basis functionsin the reference space. We will now discuss the computation of such derivatives. For the sake of clarity, werepresent in Fig. 4.3 the local ordering of the LGL points and of the associated values for a generic function q.Concerning the basis functions, for i, j, l,m = 0, . . . , k we have

∂Φij

∂ξ(ξl, ζm) = ϕ′i(ξl)ϕj(ζm) = ϕ′i(ξl)δjm

∂Φij

∂ζ(ξl, ζm) = ϕi(ξl)ϕ′j(ζm) = δilϕ

′j(ζm).

(4.26)

57


To compute the 2(k+1)4 values∂Φij

∂ξ(ξl, ζm) and

∂Φij

∂ζ(ξl, ζm) it is thus sufficient to store the (k+1)2 values

ϕ′i(ξj).

ξ

ζj

i

q0k qkk

.

.

. . ..

q02 q12 q22

q01 q11 q21

q00 q10 q20 · · · qk0

e4

e1

e2

e3

Figure 4.3.: Local ordering of the LGL points and of the associated values for a generic function q.

Let qh denote a generic problem unknown. Then we have

∂qh∂ξ

(ξi, ζj) =k∑

l,m=0

qlm∂Φlm

∂ξ(ξi, ζj) =

k∑l=0

qljϕ′l(ξi)

∂qh∂ζ

(ξi, ζj) =k∑

l,m=0

qlm∂Φlm

∂ζ(ξi, ζj) =

k∑m=0

qimϕ′m(ζj).

(4.27)

For practical implementation, we define the two arrays

q = [qij ] =

q00 q01 . . . q0k

q10 q11...

. . .qk0 qkk

, dpsi = [ϕ′i(ξj)] =

ϕ′0(ξ0) ϕ′0(ξ1) . . . ϕ′0(ξk)ϕ′1(ξ0) ϕ′1(ξ1)

.... . .

ϕ′k(ξ0) ϕ′k(ξk)

(4.28)

and compute the derivatives dqdxi =[∂qh∂ξ

(ξi, ζj)]

and dqdzeta =[∂qh∂ζ

(ξi, ζj)]

as

dqdxi = dpsiT · q, dqdzeta = q · dpsi.

To complete the description of the implementation of the high–order DG scheme, we have to consider terms ofthe form

∫K

(∂ΦI

∂xFx +

∂ΦI

∂zFz

)dx

58


as in (4.23). The previous integral can be expanded as follows:

∫K

(∂ΦK,ij

∂xFx +

∂ΦK,ij

∂zFz

)dx

=k∑

l,m=0

((∂ξ

∂x

)K,lm

∂Φij

∂ξ(ξl, ζm) +

(∂ζ

∂x

)K,lm

∂Φij

∂ζ(ξl, ζm)

)FxK,lm

wK,lm

+

((∂ξ

∂z

)K,lm

∂Φij

∂ξ(ξl, ζm) +

(∂ζ

∂z

)K,lm

∂Φij

∂ζ(ξl, ζm)

)FzK,lm

wK,lm

=k∑

l=0

wK,lj ϕ′i(ξl)

(∂ξ

∂xFx +

∂ξ

∂zFz

)K,lj

+k∑

m=0

wK,im ϕ′j(ζm)(∂ζ

∂xFx +

∂ζ

∂zFz

)K,im

Defining for each element K ∈ Th the two arrays

XFK =

[wK,ij

(∂ξ

∂xFx +

∂ξ

∂zFz

)K,ij

], ZFK =

[wK,ij

(∂ζ

∂xFx +

∂ζ

∂zFz

)K,ij

],

we compute IFK =[∫

K

(∂ΦK,ij

∂xFx +

∂ΦK,ij

∂zFz

)dx]

as

IFK = dpsi · XFK + ZFK · dpsiT .

4.7.3. Filtering the High–Frequency Modes

The DG method using high–order basis functions can be regarded as a spectral element method with no conti-nuity constraint among neighboring elements. Since high–order methods do not present any intrinsic numericaldiffusion, they are prone to instabilities due to nonlinear mixing and the Gibbs phenomenon, particularly in thecase of poorly resolved flows (see [Boy01]). The usual way of dealing with this instability in the context ofspectral formulations is to introduce a filtering operator which damps the high frequency modes without alter-ing the low frequency modes. This is done by transforming from nodal representation to modal representation,applying a low–pass filter and then by transforming back to nodal representation.

The filtering approach pursued in the present thesis is a suitable modification of the procedure proposedin [GHW02]. In this latter reference, filtering is applied with an operator splitting approach in such a waythat, denoting by Mfilter the matrix action of the filter and by q∗ the computed unfiltered solution at time leveltn+1, respectively, we have

qn+1 = Mfilter q∗. (4.29)

In our case, however, numerical experiments indicate that a straightforward use of (4.29) can trigger instabilitiesin the method when large time–steps are adopted, as allowed by the SI time integration scheme. To avoid thisproblem, we have decided to include the filtering procedure within the semi–implicit solution algorithm throughthe introduction of an additional damping operator in the continuous problem, affecting the high frequencymodes of the solution. In practice, this amounts to suitably modifying the definition of the operator L 0 in theSI splitting (4.8)–(4.9).

59


4.8. Static Condensation of the Momentum Variables byMass–Lumping

The linear system arising from the SI time discretization is usually dealt with by properly combining thecontinuity, momentum and energy equations, in such a way to obtain an algebraically equivalent problemof diffusion–reaction type (also referred to as “pseudo–Helmholtz operator”) for the sole pressure variable.This equivalent reformulation has the computational advantage of reducing considerably the dimension of theproblem, and, if an iterative solver is adopted, it produces a significant acceleration of the convergence rateand simplifies the definition of the stopping criterion. Based on these considerations, an extension of the aboveapproach to the present semi–Implicit DG setting is highly desirable, albeit being far from trivial. Nevertheless,such an extension can be made possible by conveniently exploiting the structure of the approximate quadraturerules (4.16) and (4.17). By doing so, one obtains a formulation that can be regarded as a LDG discretization ofa diffusion–reaction problem for the pressure variable where the auxiliary flux unknown is statically condensedout by proper use of mass lumping. In this sense, the resulting discrete scheme shares some similarities withhybridized dual mixed methods where static condensation is the crucial approach to end up with a linearalgebraic problem for the sole primal variable (see [AB85] for an introduction to static condensation at thediscrete level, and [CDG, Sac06] for more recent development on this subject).

For ease of presentation, we assume throughout this section that periodic boundary conditions are prescribedand we do not include the gravity terms into the implicit part of the problem. With these assumptions, the linearproblem arising from the SI–DG formulation reads:find (ρtt, Utt,Wtt, Ett) ∈ (Vh)4 such that ∀K ∈ Th, I = 1, . . . ,N

∫K

ΦIρtt dx + α

∫K

ΦI

(∂Utt

∂x+∂Wtt

∂z

)dx

−α∫

∂K

ΦI

((FL

ρ,x − FLρ,x

)nx +

(FL

ρ,z − FLρ,z

)nz

)dσ =

∫K

ΦIρ∗ dx

∫K

ΦIUtt dx + α

∫K

ΦI∂ptt

∂xdx

−α∫

∂K

ΦI

((FL

U,x − FLU,x

)nx +

(FL

U,z − FLU,z

)nz

)dσ =

∫K

ΦIU∗ dx

∫K

ΦIWtt dx + α

∫K

ΦI∂ptt

∂zdx

−α∫

∂K

ΦI

((FL

W,x − FLW,x

)nx +

(FL

W,z − FLW,z

)nz

)dσ =

∫K

ΦIW∗ dx

∫K

ΦIEtt dx + α

∫K

ΦIhh

(∂Utt

∂x+∂Wtt

∂z

)dx

−α∫

∂K

ΦI

((FL

E,x − FLE,x

)nx +

(FL

E,z − FLE,z

)nz

)dσ =

∫K

ΦIE∗ dx

∫K

ΦIptt dx =R

cv

∫K

ΦIEtt dx

(4.30)

with α = ρ−1γ∆t (see Sect. 1.2.2). Equations (4.30)2,3,4,5 then immediately provide the problem:

60

4.8. Static Condensation of the Momentum Variables by Mass–Lumping

find (ptt, Utt,Wtt) ∈ (Vh)3 such that ∀K ∈ Th, I = 1, . . . ,N∫K

ΦIptt dx + αR

cv

∫K

ΦIhh

(∂Utt

∂x+∂Wtt

∂z

)dx

−α∫

∂K

ΦI

((FL

p,x − FLp,x

)nx +

(FL

p,z − FLp,z

)nz

)dσ =

∫K

ΦIp∗ dx

∫K

ΦIUtt dx + α

∫K

ΦI∂ptt

∂xdx

−α∫

∂K

ΦI

((FL

U,x − FLU,x

)nx +

(FL

U,z − FLU,z

)nz

)dσ =

∫K

ΦIU∗ dx

∫K

ΦIWtt dx + α

∫K

ΦI∂ptt

∂zdx

−α∫

∂K

ΦI

((FL

W,x − FLW,x

)nx +

(FL

W,z − FLW,z

)nz

)dσ =

∫K

ΦIW∗ dx

(4.31)

where FLp,· = R

cvFL

E,·, where FLp,· = R

cvFL

E,· and p∗ = RcvE∗. Problem (4.31) can be fully regarded as the LDG

discretization of the following elliptic problem for the sole pressure variable

− α2a∆ p+ p = −αa(∂U∗

∂x+∂W ∗

∂z

)+ p∗ (4.32)

where a = Rcvh, supplied with the following definition of the numerical fluxes:

V = Vtt+|λ|2

cv

Rhh

[[ptt]]

p = ptt I +|λ|2

[[[Vtt]]].

(4.33)

In (4.33), for rh ∈ (Vh)2 we let

[[[rh]]] = ne,∂K ⊗ rh(ξint(K)) + ne,∂K′ ⊗ rh(ξint(K′))

(see Sect. 2.1 for the notation details). Relation (4.33)1 has the same form as (2.18)1, while (4.33)2 is amodification of (4.33)2. Notice that [[rh]] = Tr ([[[rh]]]).

Remark 4.8.1 The elliptic numerical fluxes (4.33) directly emanate from the hyperbolic numerical flux (4.19),and the dissipative term in this latter flux gives rise to the two stabilization terms in the former fluxes.

The choice (4.33) for the numerical fluxes has the same drawbacks as (2.18) in the case C22 6= 0, i.e. it isnot possible to compute Vtt element by element in terms of ptt. This implies that, starting from (4.31), it isnot possible to obtain in an efficient way a discrete counterpart of (4.32) involving the sole pressure variable.However, we show now how, by taking advantage of the approximate quadrature rule, it is possible to computeVtt node by node in terms of ptt. To this end, we need to work out the matrix formulation of (4.31), andsome additional notation is required. We denote by M the number of quadrature nodes located on Eh. Forthe Qth of such nodes, we denote by Iq(Q) the degrees of freedom collocated at the quadrature node itself,with q = 1, . . . , nQ. On the one hand, since periodic boundary conditions are assumed, we have nQ ≥ 2, onthe other hand, the regularity of Th implies an upper bound for nQ. Simmetrically, for a degree of freedom Icollocated on the element boundary, we denote by Q(I) the corresponding quadrature node. Notice that, for acontinuous function φ, we have φIq1 (Q) = φIq2 (Q) = φQ. Finally, for a given pair (Q, e), we use the shorthandnotation IQ,e, I ′Q,e to indicate two degrees of freedom such that IQ,e 6= I ′Q,e, Q(IQ,e) = Q(I ′Q,e) and IQ,e and

61


I ′Q,e belong to a couple of elements K, K ′ such that ∂K ∩ ∂K ′ = e. Notice that the subscript (Q, e) will beusually omitted, since it is clear from the context.

The integrals on the interior of K can be easily expressed in terms of the N × N matrices M , Dx and Dz

defined asMIJ =

∫K

ΨIΨJ , (Dx)IJ =∫

K

ΨI∂ΨJ

∂x, (Dz)IJ =

∫K

ΨI∂ΨJ

∂z, (4.34)

which, due to discontinuous nature of the basis functions, are block–diagonal. Moreover, thanks to the chosenbasis functions and quadrature rule, we have

MIJ = wIδIJ . (4.35)

Concerning now the boundary integrals, we illustrate the treatment of (4.31)1, as analogous considerationsapply to (4.31)2,3. Dropping the subscript tt for semplicity, we have∫

∂K

ΦI

((FL

p,x − FLp,x

)nx +

(FL

p,z − FLp,z

)nz

)dσ

=∑e∈EI

ωeI

−|λI |

2(pI − pI′) +

12RhI

cv[(UI − UI′)nx + (WI −WI′)nz]

= −

|λQ(I)|2

∑e∈EI

ωeQ(I) (pI − pI′) +

12RhQ(I)

cv

∑e∈EI

ωeQ(I) [(UI − UI′)nx + (WI −WI′)nz] .

Let now Ds, NxDs and NzD

s denote the N ×N matrices such that

(Dsq)I =12

∑e∈EI

ωeQ(I) (qI − qI′) , (Nx,zD

sq)I =12

∑e∈EI

ωeQ(I) (qI − qI′)nx,z

with q denoting either p, U or W . It is easy to verify that, up to a permutation of the unknowns, Ds, NxDs and

NzDs have a block diagonal structure withM non–zero blocks of dimension nQ,Q = 1, . . . ,M, respectively.

In particular, for the case of a quadrature node belonging to one sole edge e, we have nQ = 2 and the 2 × 2blocks

DsQ =

12

[1 −1−1 1

]ωe

Q,

and

NxDsQ =

12

[nxe,I1(Q) −nxe,I1(Q)

nxe,I2(Q) −nxe,I2(Q)

]ωe

Q, NzDsQ =

12

[nze,I1(Q) −nze,I1(Q)

nze,I2(Q) −nze,I2(Q)

]ωe

Q.

Summarizing, the matrix counterpart of (4.31) after proper use of numerical integration reads:(M + αΛDs) p+ αA [(Dx −NxD

s)U + (Dz −NzDs)W ] = Mp∗

(M + αΛDs)U + α (Dx −NxDs) p = MU∗

(M + αΛDs)W + α (Dz −NzDs) p = MW ∗

(4.36)

where Λ and A are N ×N diagonal matrices defined as

ΛIJ = |λQ(I)|δIJ , AIJ =R

cvhQ(I)δIJ .

Remark 4.8.2 Problem (4.36) can be regarded as the Generalized Galerkin version of (4.31), since exactintegration has been substituted by numerical integration. The analysis of this problem would thus require theuse of the Strang Lemma (see [QV94], Theorem 5.5.1).

62

4.8. Static Condensation of the Momentum Variables by Mass–Lumping

The key element for the reformulation of (4.36) in terms of the sole unknown p relies on an efficient computa-tion of the inverse of the matrix MDG = M + αΛDs. This can be obtained as follows:(

MDG)−1 = (M + αΛDs)−1

=(I + αΛM−1Ds

)−1M−1 = ΣM−1.

(4.37)

Since Σ−1 =(I + αΛM−1Ds

), it has the same block diagonal structure asDs, and its computation is straight-

forward (see Fig. 4.4).

Q1

σQ4

11σ

Q4

12σ

Q4

13σ

Q4

14

σQ4

21σ

Q4

22σ

Q4

23σ

Q4

24

σQ4

31σ

Q4

32σ

Q4

33σ

Q4

34

σQ4

41σ

Q4

42σ

Q4

43σ

Q4

44

σQ3

11σ

Q3

12σ

Q3

13

σQ3

21σ

Q3

22σ

Q3

23

σQ3

31σ

Q3

32σ

Q3

33

σQ2

11σ

Q2

12

σQ2

21σ

Q2

22

[

σQ1

11

]

Q4

Q2

Q3

Figure 4.4.: Representation of the block diagonal structure of the matrix Σ. A 1 × 1 block is associated withinternal quadrature nodes (yellow) such asQ1, and we have σQ1

11 = 1. Quadrature nodes belongingto one sole edge (green), such as Q2, originate 2 × 2 blocks. Finally, nodes belonging to two ormore edges (blue and red), such as Q3 and Q4, originate nQ × nQ blocks.

Upon defining DDGx =

(MDG

)−1 (Dx −NxDs), DDG

z =(MDG

)−1 (Dz −NzDs), p = Σp∗, U = ΣU∗

and W = ΣW ∗, system (4.36) can be written asp+ αA

[DDG

x U + DDGz W

]= p

U + αDDGx p = U

W + αDDGz p = W .

(4.38)

Notice that, since hh is continuous across interelement boundaries, we have that ΣA = AΣ. Substitut-ing (4.38)2,3 into (4.38)1 we obtain

p− α2A

[(DDG

x

)2

+(DDG

z

)2]p = p− αA

(DDG

x U + DDGz W

). (4.39)

Problem (4.39) is the discrete counterpart of (4.32). The advantages of solving of (4.39) instead of (4.36) willbe numerically demonstrated in Sect. 5.1.4.

63

Chapter 5.

Numerical Validation

In this chapter, the semi–Implicit DG method introduced in Chapt. 4 is numerically val-idated on a series of classical benchmark test cases. First, in Sect. 5.1 the warm bubbleand cold bubble nonlinear cases are considered. The aim of these tests is to show someadvantages of the proposed semi–Implicit DG formulation when compared to other wellestablished methods. Then, in Sect. 5.2 a set of test cases more closely related to mesoscaleatmospheric modeling is considered, namely, the waves which develop in a stratified flowpassing across a mountain, referred to as mountain waves. The aim of these tests is to pro-vide a physical validation of the proposed formulation, showing that the results reportedin the literature can be recovered. A critical element for these latter simulations is theintroduction of the sponge layer, as discussed in Sect. 4.3.

5.1. Bubble Convection Experiments

In this section, we consider four idealized test cases characterized by buoyancy driven flows. In these tests,a basic–state atmosphere is considered, which is assumed to be at rest and in hydrostatic equilibrium, and athermal anomaly, with a consequent density perturbation, is introduced. The resulting flow, which is inherentlynonlinear, strongly depends on the relative intensity of two competing effects: on the one hand, the vorticityproduced by the initial density gradient in conjunction with the gravitational field tends to produce arbitrar-ily small–scale structures, on the other hand, diffusion produces an homogenization of the flow, and smallscale structures are destroyed at a higher rate than large scale ones. As a consequence, for any fixed com-putational grid, only diffusion dominated flows can be well resolved for arbitrarily long time intervals, whilevanishing diffusion flows will eventually develop subgrid scale structures. To deal with such unresolved struc-tures, viscosity must be introduced in the numerical formulation, either implicitly, by considering low orderor monotonic schemes, or explicitly, by increasing the diffusion coefficient. Since the high–order DG methodconsidered in this thesis does not present intrinsic diffusion, we will follow the second approach. Concerningthen the basic–state atmosphere, it can be either isothermal or neutral; regardless of this, an isothermal atmo-sphere with temperature T equal to the highest temperature in the basic–state profile will be employed for theSI splitting of Sect. 4.2. Finally, we notice that, although all computations are performed using the conservativevariables (1.16), the results are then converted to primitive variables (1.14) to ease of comparison with the lit-erature, and the Courant number is defined as C = v∆t

hLGL , where hLGL denotes the (variable) spacing betweenthe Legengre–Gauss–Lobatto points (see Sect. 4.5.1).

Chapter 5. Numerical Validation

5.1.1. Density Current Test Case

The test considered in this section is the so–called density current test case, proposed in [SWW+93] as abenchmark test case for numerical methods for the Navier–Stokes Equations. For this case, the computationaldomain is the rectangle [−25.6 km , 25.6 km]× [0 km , 6.4 km] with reflective boundary conditions. The initialcondition is represented by a neutrally stratified atmosphere with surface temperature 300K where a thermalanomaly of amplitude −15K, centered at xc = 0 km, zc = 3 km, is introduced. Notice that, thanks tothe symmetry of the problem, computations are performed only for 0 km ≤ x ≤ 25.6 km. The diffusioncoefficient is assumed to be ν = 75m2s−1, while we refer to [SWW+93] for all the remaining details. It isimportant to notice that the diffusion coefficient is chosen in such a way that the flow is diffusion limited anda grid–converged reference solution can be generated using high spatial resolution. In practice, this is done byconsidering a grid resolution of 25m. The performance of various numerical models are then compared at thefollowing spatial resolutions: 400m, 200m, 100m and 50m.

To compare the results obtained with the semi–Implicit DG method (SIDG, in this chapter) to thosein [SWW+93], four computational grids are considered such that the number of grid points is equal to thenumber of grid points used in the reference for the four resolutions. We thus employ 8× 2, 8th–order elements(resolution 400m), 16 × 4, 8th–order elements (resolution 200m), 32 × 8, 8th–order elements (resolution100m) and 64 × 16, 8th–order elements (resolution 50m). Figure 5.1 shows the potential temperature θcomputed with the SIDG method at time level 900 s for the four computational grids, while Fig. 5.2, takenfrom [SWW+93], shows the reference solution together with the results obtained with the Full Local Spectralmethod (FLS), the Monotonic Upstream method (MUPL) and the Piecewise Parabolic Method (PPM) at thesame time level. It can be seen that at the coarsest resolution the SIDG method clearly resolves one of the threerolls present in the reference solution, yielding a result which is comparable to the low order monotonic MUPLand PPM solutions, although with the presence of some spurious oscillations. This is significant because, onthe contrary, the high order FLS method produces absolutely meaningless results. When the 200m resolutionis considered, the SIDG solution presents two fully developed rolls, thus resulting comparable to the FLS solu-tion and clearly superior to the MUPL and PPM solutions, where only one roll is reproduced. Finally, at higherresolutions, such as 100m, the high order SIDG and FLS solutions are very close to the reference solution.These results suggest that the SIDG scheme can combine spectral accuracy in the case of well resolved flowswith the capability of producing meaningful results even in the case of poorly resolved flows.

m

m

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2000

4000

m

m

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2000

4000

m

m

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2000

4000

m

m

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

2000

4000

Figure 5.1.: Density current test case, deviations from the neutral basic–state atmosphere of the potential tem-perature θ at time level 900 s. Contour intervals are 1K. The spatial resolution increases from400m (bottom, right), to 200m (top, right), to 100m (bottom, left) and 50m (top, left). Thetime steps are chosen in such a way to ensure that the maximum Courant number is 1.5 and themaximum advective Courant number is 0.15.

To verify the effect of the computational grid on the numerical solution, the test has been repeated including

66


Figure 5.2.: Density current test case, deviations from the neutral atmosphere of the potential temperature θ attime level 900 s (from [SWW+93]). Reference solution (bottom, right), and numerical solutionwith resolution 400m (top, right), 200m (bottom, left) and 100m (top, left).

a uniform horizontal flow with u = 56.89ms−1 and introducing periodic boundary conditions on the lateralboundaries. The uniform flow is chosen in such a way that time level 900 s corresponds to one period, sothat the final configuration should be equal to the one with no horizontal flow, and in particular it should besymmetric with respect to the vertical axis x = 0 km. The results obtained with resolution 100m are displayedin Fig. 5.3. It can be noted that the computed solution presents a very good symmetry and that it is notsignificantly different from the one in Fig. 5.1, thus indicating that the numerical method is robust with respectto modifications of the computational grid.

5.1.2. Robert Test Case N 1

We consider in this section the first test case presented in [Rob92]. The computational domain is the square[0m, 1000m]2 with reflective boundary conditions. The initial condition is represented by a neutrally stratifiedatmosphere with surface temperature 303K where a circular bubble with a uniform potential temperature0.5K in excess of the basic–state atmosphere is introduced. The original test described in [Rob92] does notinclude diffusion. However, when a high–order method is adopted, due to the discontinuity in the initial datum,it is impossible to compute an approximate solution unless a minimum amount of viscosity is introduced.Numerical experiments show that, for a 12 × 12, 8th–order element computational grid (which has the same

67


m

m

−1.5 −1 −0.5 0 0.5 1 1.5

x 104

0

2000

4000

m

m

−1.5 −1 −0.5 0 0.5 1 1.5

x 104

0

2000

4000

m

m

−1.5 −1 −0.5 0 0.5 1 1.5

x 104

0

2000

4000

m

m

−1.5 −1 −0.5 0 0.5 1 1.5

x 104

0

2000

4000

Figure 5.3.: Density current test case with uniform horizontal flow and periodic lateral boundary conditions,solution at time level 900 s. Deviations from the neutral atmosphere for the Exner pressure π (top)(contour intervals 1 · 10−4), the horizontal velocity u (second row) (contour intervals 2ms−1), thevertical velocity w (third row) (contour intervals 2ms−1) and the potential temperature θ (bottom)(contour intervals 1K). Resolution: 400m.

number of grid points as a uniform grid of amplitude 10.42m), the value ν = 0.2m2s−1 ensures stability ofthe high order numerical solution, so that this value will be chosen. Figure 5.4 shows the computed solution attime levels 420 s and 600 s on a 12 × 12, 8th–order element grid. These results qualitatively agree with thosereported in [Rob92] as well as and with those reported in Fig. 5.5 obtained on a refined 24 × 24, 8th–orderelement grid. This fact suggests that the viscosity explicitly introduced in the high–order scheme is of the sameorder as the numerical viscosity of the low–order scheme used by Robert. Notice also that, since the flow is notcompletely resolved on the considered grids, only a qualitative agreement should be expected.

For further comparisons, figure 5.6 represents the solution computed with an explicit in time, continuous inspace Spectral Element method (SE) on the same 12 × 12, 8th–order element grid used for the SIDG compu-tations. First of all, a good agreement between the results provided by the SE formulation and by the SIDGformulation should be observed. In addition, a closer examination reveals that the SIDG solution is less affectedby spurious oscillations. Notice that, since the potential temperature satisfies an advection–diffusion equation,values of θ′ lower than 0K or larger than 0.5K represent spurious local extrema. Table 5.1 summarizes maxi-mum and minimum values for the SE and SIDG solutions, confirming that the DG formulation is more robustin the treatment of the irregular datum.

68


m

m

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

m

m

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

Figure 5.4.: Robert test case N 1, SIDG method, deviation from the neutral basic–state atmosphere of thepotential temperature θ. The computational grid is composed by 12 × 12, 8th–order elements,while the time step is 0.2 s, yielding a maximum Courant number 19.3 and a maximum advectiveCourant number 0.13. Solution at time level 420 s (left) and 600 s (right).

θ′min,420 s θ′max,420 s θ′min,600 s θ′max,600 s

SE, 10.4m −0.085 0.581 −0.093 0.598SIDG, 10.4m −0.057 0.550 −0.053 0.561SIDG, 5.2m −0.020 0.522 −0.016 0.527

Table 5.1.: Maximum and minimum values of the potential temperature θ (values in K).

5.1.3. Robert Test Case N 2

In this section, we consider the second test case proposed in [Rob92]. This test is similar to the test describedin Sect. 5.1.2, but in the present case the profile of the thermal anomaly is smooth. In addition, the height of thecomputational domain is increased, so that the rectangle [0m, 1000m] × [0m, 1500m] is considered. In theoriginal test case, inviscid flow is considered. When employing high order methods to compute an approximatesolution, a sharp gradient develops at the top of the warm bubble approximately at time level 720 s, so that it isnot possible to continue the time integration beyond this time level unless some viscosity is introduced. In ourtests, we found that, employing a 12× 18 grid with 8th–order elements, an explicit viscosity ν = 0.1m2s−2 issufficient to dissipate subgrid scales for arbitrarily long time intervals. The computed potential temperature θ isrepresented in Fig. 5.7 for time levels 360 s, 720 s and 1080 s. For these computations, the maximum Courantnumber was 15, while the maximum advective Courant number was 0.14. These results are in good agreementwith those reported in [Rob92].

5.1.4. Cold Bubble Test Case with Static Condensation

In this section, we demonstrate experimentally the practical advantages of employing the static condensationprocedure illustrated in Sect. 4.8. We consider here a test case with periodic boundary conditions on all thedomain boundaries and, following the framework presented in Sect. 4.8, we treat implicitly only the terms as-

69


m

m

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

m

m

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

Figure 5.5.: As in Fig. 5.4 but employing a refined 24 × 24, 8th–order element grid and a correspondinglyhalved time step.

sociated with acoustic waves. Since gravity waves are treated explicitly, they pose a limitation on the maximumstable time step which is more stringent that the one associated with advection (see [Dur99]). Nevertheless, thestable time step in the SI case is still approximately 40 times larger that what would be possible in the explicitcase. The extension of the static condensation procedure to include general boundary conditions and implicittreatment of gravity waves is currently being addressed.

The computational domain is the rectangle [0m, 1000m]× [0m, 2000m], and the initial datum is representedby a thermal anomaly introduced in an isothermal atmosphere at T = 303K (notice that, thanks to this choice,the deviations from the reference atmosphere are zero far from the thermal anomaly, which allows to enforceperiodic boundary conditions on top and bottom boundaries). The thermal anomaly is given by

θ′0 =A, r ≤ r0Ae−(r−r0)

2/σ2, r > r0,

with r2 = (x − x0)2 + (z − z0)2 and x0 = 500m, z0 = 1250m, r0 = 50m, σ = 100m and A = −15K.Viscosity is set equal to 0.2m2s−1. The computational grid is composed by 12× 24, 16th–order elements, sothat the number of grid points is the same as that of a uniform grid with resolution 5.2m, while the time stepis 0.02 s, corresponding to a maximum Courant number 7.16 and a maximum advective Courant number 0.14.The computed potential temperature θ is displayed in Fig. 5.8 for time levels 50 s, 100 s, 150 s and 200 s.

A comparison between the solutions computed with and without performing the static condensation operationshows that the number of GMRES iterations required for the solution of the linear system decreases by ap-proximately a factor 3 (from 45 to 15 iterations), while a rough comparison with the explicit time integrationindicates a reduction in the overall computational time of approximately a factor 5. Here, we point out thatthese efficiency considerations need further verification on a wider set of test cases; nevertheless, the greaterefficiency of the SIDG with mass lumping and static condensation seems to be out of question.

70

5.2. Mountain Waves Approximation

m

m

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

m

m

0 200 400 600 800 10000

100

200

300

400

500

600

700

800

900

1000

Figure 5.6.: As in Fig. 5.4 but employing an explicit in time, continuous in space Spectral Element formulation.


The Earth atmosphere is exceedingly sensitive to vertical motion, mainly for two reasons. First, the strongstable stratification gives the atmosphere a resistance to vertical displacement. Buoyancy forces, in fact, tendto return vertically displaced air parcels to their equilibrium level and this restoration is capable of originatingbroad horizontal excursion and strong winds. Second, the lower atmosphere is usually so rich in water vaporthat slight adiabatic ascent will bring the air to saturation, leading to condensation and possibly precipitation.Since a common source of vertical motion is the terrain orography, the study of airflow past mountains hasreceived great attention in the meteorological community, both from the analytical and numerical viewpoints.A classical review of the analytical results available for the mountain wave problem is [Smi79], and a compre-hensive description of the main features of the air motion in a stratified atmosphere can be found in [Tol63].Benchmark test cases for hydrostatic and non hydrostatic numerical models based on the simulation of moun-tain waves are proposed in [KL78, KD83, DK83, PBRL95, Bon00].

5.2.1. Model Setup

In all the numerical experiments of the present section, a constant stability background atmosphere is con-sidered, i.e. the Brunt–Vaisala frequency N is assumed to be strictly positive and uniform. Thanks to thehypothesis of hydrostatic balance, the vertical profile of all the thermodynamical variables is completely deter-mined once N and the surface temperature T0 are specified. In particular, the temperature T is given by

T = T0

[1 + ∆

(e

N2g z − 1

)](5.1)

where ∆ = 1− g2

cpN2T0. It can be noted that:

• if ∆ < 0, i.e. T0 <g2

cpN2 , the temperature decreases with height;

71


Figure 5.7.: Robert test case N 2, deviation from the neutral basic–state atmosphere of the potential temper-ature θ. The computational grid is composed by 12 × 18, 8th–order elements, while the time stepis 0.1 s, yileding a maximum Courant number 15 and a maximum advective Courant number 0.14.Initial condition (top, left) and solution at time levels 360 s (bottom, left), 720 s (top, right) and1080 s (bottom, right).

• if ∆ = 0, i.e. T0 = g2

cpN2 , the atmosphere is isothermal;

• if ∆ > 0, i.e. T0 >g2

cpN2 , the temperature increases with height.

The forcing term is represented by the terrain orography. A single versiera of Agnesi mountain profile isintroduced

h(x) =hm

1 +(

xa

)2 ,hm and a denoting the mountain height and half width, respectively. The initial and boundary conditions arerepresented by a uniform horizontal flow with velocity u and zero deviation from the background atmosphere.

The theoretical analysis presented in [Smi79] indicates that in the steady state configuration the flow is char-acterized by the vertical wavelength λz = 2π u

N ; moreover, the characteristic time for this problem is given byt = a

u . Three very different flow regimes are possible, depending on the ratio between N and the characteristicfrequency f = t−1. On the one hand, when narrow mountains and week stability are considered, f N and

72


the internal gravity waves are not excited by the obstacle. In this case, the stratification has virtually no effectand the flow is qualitatively similar to irrotational flow, since the disturbance decays with height. This flow isinherently nonhydrostatic as the vertical pressure gradient in (4.4)3 is equilibrated by the vertical acceleration.On the other hand, when wide mountains and strong stability are considered, f N and the obstacle generatesinternal gravity waves. These waves propagate vertically and thus the disturbance does not decay upward. Thevertical pressure gradient in (4.4)3 is roughly equilibrated by the density gradient, and the vertical accelerationis negligible. The flow is thus nearly hydrostatic and, in this case, vorticity is produced by the density gradient.Finally, when f ≈ N , the flow is neither irrotational nor hydrostatic. Since gravity waves are still emitted, thedisturbance does not decay with height, vertical accelerations are not negligible, though, and the wave patternis different from the hydrostatic case.

All the test cases consider inviscid flow, so that in practice the Euler Equation are solved. Although the compu-tations are performed employing the conservative variables as in (1.16), for ease of comparison with the resultsin the literature the results are converted into the primitive variables π, u, w and θ defined in (1.15). In additionto the plots of the primitive variables, the normalized vertical fluxes of horizontal momentum are reported,which are defined as follows: the vertical flux of horizontal momentum at a generic height z is defined as

m(z) =∫ +∞

−∞ρ(x, z)u(x, z)w(x, z) dx

and the normalized flux is computed as m(z)/m, with

m =π

4ρ0Nuh

2m

being the theoretical vertical flux of horizontal momentum in the linear hydrostatic case (see [Smi79]).

A critical element in the simulation of mountain waves is represented by the sponge layer, as discussed inSect. 4.3, which is specified through the damping coefficients τπ and τ in (4.7). In practice, two dampingprofiles are specified for the upper boundary and for the lateral boundaries. Concerning the upper boundary, areflection of the vertically propagating internal gravity waves would significantly perturb the steady state waveprofile, and must be avoided. In this region, we set τπ = 0 and τ = τg + τa, with

τg(z) =

0, for z ≤ zD

α2

(1− cos

(z−zD

zT−zDπ))

, for 0 ≤ z−zD

zT−zD≤ 1

2

α2

(1 +

(z−zD

zT−zD− 1

2

)π), for 1

2 ≤z−zD

zT−zD≤ 1

where zD and zT denotes the top of the undamped region and of the computational domain, respectively, andα is a constant coefficient, and

τa(z) =1− tanh

(zT−z

dz

)tanh

(zT−z

dz

)where dz is a characteristic length. The term τg is used to damp the internal gravity waves, and the coefficientsα and zD can be computed following [KL78] as follows: zD = zT − 3

2λz and α = 6π uλx

, where the dominanthorizontal wavelength λx is assumed to be equal to 2a. In practice, τg is non zero in a non negligible part ofthe computational domain, from one third to one half. The term τa is introduced to damp the short acousticwaves which are emanated from the obstacle in the very beginning of the simulation. The characteristic lengthdz is much smaller that the quantity zT − zD, and τa is negligible for z ≤ zT − 3dz . Notice that, as discussedin Sect. 1.2.3, the fact that τa →∞ for z → zT is handled naturally in the discretized problem.

Concerning the lateral boundaries, our experience indicates that they are not as critical as the upper boundary,and various different damping profiles can be considered with equally good results. For the computations

73


presented in this thesis, we have set τπ = τ and

τ(x) =1− tanh

(|xS−x|

dx

)tanh

(|xS−x|

dx

)where xS denotes the abscissa of the closest lateral boundary and dx is a characteristic distance.

For all the test cases, the reference profile employed for the SI splitting (1.18) is isothermal, and the temperatureT is chosen as the maximum of the background profile (5.1), as suggested in [SHB78]. Notice that this canlead to deviations of the computed solution from the reference state up to 30%, without affecting the stabilityof the SI time marching schemes. Time integration is performed with the BDF2a scheme in Tab. 1.1, althoughsimilar results are obtained when LF2 or BDF2b are employed.

5.2.2. Linear Hydrostatic Flow

For this test case, we set N = 0.0179 s−1 and T0 = 300K, so that the temperature profile is isothermal.The mountain is characterized by a = 10 km, hm = 1m, and we set u = 20ms−1. This yields f =0.002 s−1, so that the steady state flow is hydrostatic. Since the height of the mountain can be mathematicallyconsidered as an infinitesimal quantity, the problem can be linearized and a closed for analytic solution isavailable (see [Smi79]). The computational domain is 240 km wide and 24 km high, the final time is tfin =12h, corresponding to 86 t, and the absorbing layer is defined by zD = 11.5 km, α = 0.02s−1, dz = 0.66 kmand dx = 8 km. The computational grid is composed by 10 × 10, 8th–order elements, so that the number ofgrid point is the same as in the case of a uniform grid with ∆x = 3 km and ∆z = 300m, while the time stepis ∆t = 7 s, corresponding to a maximum acoustic Courant number 23 and a maximum advective Courantnumber 0.12.

Fig. 5.9 represents the contour lines of the computed (black, continuous line) and analytic (red, dashed line)solution. It can be seen that a satisfactory agreement is achieved, and is comparable with other results reportedin the literature. Fig. 5.10 represents the normalized vertical flux of horizontal momentum as a function of thevertical coordinate, for various time levels (the ripples are due to the interpolation of the numerical solutionat the various vertical levels z). In the undamped region, the computed value is close to the theoretical value1. A comparison between the curves represented indicates that the asymptotic steady state configuration is notreached yet, and small adjustment of the numerical solution is still taking place.

5.2.3. Linear Nonhydrostatic Flow

For this test case, we set N = 0.01 s−1 and T0 = 280K, so that temperature decreases with height. Themountain is characterized by a = 1 km, hm = 1m, and we set u = 10ms−1. This yields f = 0.01 s−1,so that the flow is neither hydrostatic nor irrotational. For convenience, we refer to the present regime as“nonhydrostatic”. Since the height of the mountain is again infinitesimal, the problem can be linearized anda solution can be obtained by means of Fourier transform techniques. Such a solution, however, can not beeasily represented in closed form. The computational domain is 144 km wide and 30 km high, the final timeis tfin = 5h, corresponding to 180 t, and the absorbing layer is defined by zD = 16 km, α = 0.09s−1,dz = 0.66 km and dx = 8 km. The computational grid is composed by 20×10, 8th–order elements, so that thenumber of grid point is the same as in the case of a uniform grid with ∆x = 900m and ∆z = 380m, whilethe time step is ∆t = 6 s, corresponding to a maximum acoustic Courant number 16 and a maximum advectiveCourant number 0.16.

74


Fig. 5.11 represents the contour lines of the computed solution. These results compare well with analogous testsdescribed in the literature. It can be seen, however, that some reflection is still present in spite of the absorbinglayer. Further tuning of the damping coefficient τg is thus required in order to minimize this effect. Fig. 5.12represents the normalized vertical flux of horizontal momentum as a function of the vertical coordinate, forvarious time levels (the ripples are again due to the interpolation of the numerical solution at various verticallevels z). For this test case, a theoretical value of the normalized flux can be computed as in [KD83], Eq. (37),yielding 0.456, which is in good agreement with the numerical result.

5.2.4. Linear Irrotational Flow

For this test case, we set N = 0.008 s−1 and T0 = 280K, so that temperature decreases with height. Themountain is characterized by a = 500m, hm = 1m, and we set u = 40ms−1. This yields f = 0.08 s−1, sothat the flow is irrotational. Since the height of the mountain is, as in the previous cases, an infinitesimal quan-tity, the problem can be linearized and the solution is known in closed form for the variable w (see [Smi79]).The computational domain is 20 km wide and 6.5 km high, the final time is tfin = 20min, correspondingto 87 t and the absorbing layer is defined by dz = 0.66 km and dx = 2.67 km. Notice that, since no gravitywaves are generated, we set α = 0 s−1. The computational grid is composed by 20 × 14, 8th–order elements,so that the number of grid point is the same as in the case of a uniform grid with ∆x = 130m and ∆z = 60m,while the time step is ∆t = 0.2 s, corresponding to a maximum acoustic Courant number 3.6 and a maximumadvective Courant number 0.16.

Fig. 5.13 represents the contour lines of the computed solution. It can be verified that the computed verticalvelocity is in good agreement with the analytical solution, and that the remaining fields are similar to thosereported in the literature. Fig. 5.14 represents the contour lines for the potential temperature θ, computed bysumming the reference value θ and the deviation θ′, this latter amplified by a factor 1000. Since the flow isinviscid, constant potential temperature lines coincide with the flow streamlines. It can be seen that the flow issymmetric across the obstacle. As a consequence, there is no drag and the vertical flux of horizontal momentumis zero. Moreover, it can be seen that in this case the disturbance decays with height.

5.2.5. Nonlinear Hydrostatic Flow

For this test case, we set N = 0.02 s−1 and T0 = 273K, so that temperature increases with height. Themountain is characterized by a = 16 km, hm = 800m, and we set u = 32ms−1. This yields f = 0.002 s−1,so that the steady state flow is hydrostatic. The finite height of the mountain makes the nonlinear effects nonnegligible. The computational domain is 500 km wide and 28.5 km high, the final time is tfin = 22.5h,corresponding to 162 t, and the absorbing layer is defined by zD = 11.5 km, α = 0.02s−1, dz = 0.66 kmand dx = 4 km. The computational grid is composed by 23 × 22, 8th–order elements, so that the number ofgrid point is the same as in the case of a uniform grid with ∆x = 2.72 km and ∆z = 160m, while the timestep is ∆t = 2 s, corresponding to a maximum acoustic Courant number 12 and a maximum advective Courantnumber 0.06.

Fig. 5.15 represents the contour lines of the computed solution. These results are similar to those obtained foranalogous test cases in the literature. Fig. 5.16 represents the contour lines for the potential temperature θ,computed by summing the reference value θ and the deviation θ′. Since the flow is inviscid, constant potentialtemperature lines coincide with the flow streamlines. The flow field can be compared with the analytic solutionin the case of a circular obstacle obtained in [HM69]. Finally, Fig. 5.17 represents the normalized vertical fluxof horizontal momentum as a function of the vertical coordinate, for various time levels. For this test case, afirst order analytic solution presented in [MH69] (see Tab. 2) provides the value 1.11. It can be seen that at

75


the time level 16.2 t the computed momentum flux is very close to this value, however when the steady stateconfiguration is attained the normalized momentum flux reaches the value 1.5. Further investigation is requiredon this discrepancy, however we notice here that our results are compatible with those presented in [KD83],Fig. 7, where it is shown that the solution in the nonlinear case is sensitive to the position zD of the spongelayer.

5.2.6. Nonlinear Nonhydrostatic Flow

For this test case, we set N = 0.02 s−1 and T0 = 273K, so that temperature increases with height. Themountain is characterized by a = 1 km, hm = 450m, and we set u = 13.28ms−1. This yields f =0.013 s−1, so that the regime is nonhydrostatic. The finite height of the mountain makes the nonlinear effectsnon negligible. The computational domain is 40 km wide and 20 km high, the final time is tfin = 5h,corresponding to 240 t, and the absorbing layer is defined by zD = 9 km, α = 0.13s−1, dz = 0.66 km anddx = 2 km. The computational grid is composed by 25 × 25, 8th–order elements, so that the number of gridpoint is the same as in the case of a uniform grid with ∆x = 200m and ∆z = 100m, while the time step is∆t = 0.5 s, corresponding to a maximum acoustic Courant number 5.25 and a maximum advective Courantnumber 0.09.

Fig. 5.18 represents the contour lines of the computed solution, while Fig. 5.19 represents the contour linesfor the potential temperature θ. In general the flow appears to be well resolved, and the results are compatiblewith those in the literature. However, the decay of the generated wave downstream the obstacle is slower thanreported in other works, such as [Bon00]. With this respect, it should be noted that this test is exceedinglysensitive to the position of the absorbing layer. Figure 5.20 represents the normalized vertical flux of horizontalmomentum as a function of the vertical coordinate, for various time levels. On the one hand, the computedvalue is in good agreement with the analytic value 0.65, on the other hand, this plot indicates that steady stateconditions are not reached yet.

The nonhydrostatic nonlinear test is then repeated with a higher mountain, with hm = 900m. In this casewe observe a breaking of the wave generated by the obstacle, and a shed of downwind propagating vortexes.Figure 5.21 represents the contour lines of the computed u and w variables and the velocity vector at the timelevel 3h, corresponding to 143 t. This test demonstrates the robustness of the SIDG method in the case ofcomplex unsteady flows.

76


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Figure 5.8.: Cold bubble in an isothermal atmosphere, deviation from the basic–state atmosphere of the poten-tial temperature θ at time levels 50 s (top, left), 100 s (bottom, left), 150 s (top, right) and 200 s(bottom, right). The computational grid is composed by 12 × 24, 16th order elements while thetime step is 0.02 s.

77


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Figure 5.9.: Linear hydrostatic isothermal case at t = 86t, deviations from the mean flow: computed solu-tion (black continuous line) and analitic solution (red dashed line). Top, left: Exner pressure π,values between −1.5 · 10−6 and 1.5 · 10−6, with contour interval of 2.14 · 10−7; top, right: hor-izontal velocity u, values between −3.0 · 10−2ms−1 and 3.0 · 10−2ms−1, with contour intervalof 4.29 · 10−3ms−1; bottom, left: vertical velocity w, values between −4.0 · 10−3ms−1 and4.0 · 10−3ms−1, with contour interval of 5.71 · 10−4ms−1; bottom, right: potential temperatureθ values between −2.5 · 10−2K and 2.5 · 10−2K, with contour interval of 3.57 · 10−3K.

78


0 0.2 0.4 0.6 0.8 1 1.2 1.40

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8.6 t25.8 t43.0 t60.2 t77.4 t86.0 t

8.6 t

25.8 t

43.0 t

60.2 t

77.4 t

86.0 t

Figure 5.10.: Linear hydrostatic isothermal case, with time ranging from 8.6t to 86t, normalized vertical fluxof horizontal momentum.

79


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Figure 5.11.: Linear nonhydrostatic constant stability case at t = 180t, deviations from the mean flow. Top,left: Exner pressure π, values between −4 · 10−7 and 4 · 10−7, with contour interval of 5.71 ·10−8; top, right: horizontal velocity u, values between −1.0 · 10−2ms−1 and 1.0 · 10−2ms−1,with contour interval of 1.43 · 10−3ms−1; bottom, left: vertical velocity w, values between−5.5 ·10−3ms−1 and 5.5 ·10−3ms−1, with contour interval of 7.85 ·10−4ms−1; bottom, right:potential temperature θ values between −3.6 · 10−3K and 3.6 · 10−3K, with contour interval of5.14 · 10−4K.

80


0 0.1 0.2 0.3 0.4 0.5 0.60

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18 t 54 t 90 t126 t162 t180 t

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180 t

Figure 5.12.: Linear nonhydrostatic constant stability case, with time ranging from 18t to 180t, normalizedvertical flux of horizontal momentum.

81


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Figure 5.13.: Linear irrotational case at t = 87t, deviations from the mean flow: computed solution (black con-tinuous line) and analitic solution (red dashed line). Top, left: Exner pressure π, values between−1.2 · 10−5 and 1.7 · 10−6, with contour interval of 9.79 · 10−7; top, right: horizontal velocity u,values between−8.0·10−3ms−1 and 8.5·10−2ms−1, with contour interval of 6.64·10−3ms−1;bottom, left: vertical velocity w, values between −5.2 · 10−2ms−1 and 5.2 · 10−2ms−1, withcontour interval of 7.43 · 10−3ms−1; bottom, right: potential temperature θ values between−1.8 · 10−3K and 1.0 · 10−4K, with contour interval of 1.36 · 10−4K.

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Figure 5.14.: Linear irrotational case at time 87 t, constant potential temperature surfaces, coinciding with theflow trajectories. Notice that the vertical displacement have been amplified by a factor 1000 forvisualization purposes.

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Figure 5.15.: Nonlinear hydrostatic case at t = 162 t, deviations from the mean flow. Top, left: Exner pressureπ, values between −3.1 · 10−3 and 1.2 · 10−3, with contour interval of 3.07 · 10−4; top, right:horizontal velocity u, values between −2.3 · 101ms−1 and 2.8 · 101ms−1, with contour intervalof 3.74ms−1; bottom, left: vertical velocity w, values between −3.9ms−1 and 3.5ms−1, withcontour interval of 5.28 · 10−1ms−1; bottom, right: potential temperature θ values between−2.2 · 101K and 2.7 · 101K, with contour interval of 3.5K.

84


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Figure 5.16.: Nonlinear hydrostatic case at time 162 t, constant potential temperature surfaces, coinciding withthe flow trajectories.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

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48.6 t

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162.0 t

Figure 5.17.: Nonlinear hydrostatic case, with time ranging from 16.2 t to 162 t, normalized vertical flux ofhorizontal momentum.

85


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Figure 5.18.: Nonlinear nonhydrostatic case at t = 240 t, deviations from the mean flow. Top, left: Exnerpressure π, values between −4.2 · 10−4 and 2.0 · 10−4, with contour interval of 4.43 · 10−5;top, right: horizontal velocity u, values between −7.2ms−1 and 9.0ms−1, with contour intervalof 1.16ms−1; bottom, left: vertical velocity w, values between −4.2ms−1 and 4.0ms−1, withcontour interval of 5.86 · 10−1ms−1; bottom, right: potential temperature θ values between−6.0K and 9.0K, with contour interval of 1.07K.

86


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Figure 5.19.: Nonlinear nonhydrostatic case at time 240 t, constant potential temperature surfaces, coincidingwith the flow trajectories.

−0.2 0 0.2 0.4 0.6 0.8 1 1.20

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data1data2data3data4data5data6

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Figure 5.20.: Nonlinear nonhydrostatic case, with time ranging from 24.0 t to 240 t, normalized vertical flux ofhorizontal momentum.

87


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Figure 5.21.: Nonlinear nonhydrostatic case with breaking wave at t = 143 t. Top, left: horizontal velocity u,values between −2.2 · 101ms−1 and 3.0 · 101ms−1, with contour interval of 3.71ms−1; right:vertical velocity w, values between −1.3 · 101ms−1 and 1.0 · 101ms−1, with contour interval of1.68ms−1; bottom: velocity vector.

88

Chapter 6.

Conclusions and Future Work

The main purpose of the present thesis is to devise robust, accurate and computationally efficient numericaltechniques for the discretization of partial differential models in environmental fluid dynamics. With this aim,the following novel approaches are proposed and validated on several benchmark test cases:

• a Semi–Lagrangian Discontinuous Galerkin method (SLDG) for the linear advection–diffusion equation;

• a Semi–Implicit Discontinuous Galerkin method (SIDG) for the Navier–Stokes Equations for a stratifiedfluid.

The main results obtained in the analysis are here summarized for both proposed formulations.

Concerning the SLDG method, we mention that:

• the scheme, which is the first attempt to combine SL and DG discretizations, is derived for arbitrarilyhigh–order elements. Moreover, it is proven to be stable independently of the Courant number and to beoptimal in many respects for applications to large advection–reaction systems;

• a monotonization technique is derived for a generic DG scheme, based on the Flux Corrected Transportapproach, that greatly reduces numerical diffusion of standard monotonization methods currently in usefor the DG method. This allows to prove a discrete maximum principle for the SLDG formulation, atleast in the case of incompressible flows.

Concerning the SIDG method, we mention that:

• for the first time, a high–order DG discretization approach is employed for the solution of the Navier–Stokes Equations for a variable density, compressible fluid under the action of gravity, in the context ofatmospheric flows;

• a general Semi–Implicit time discretization approach is derived for this spatial discretization, along thesame lines as in traditional SI approaches for finite difference and low order finite element methodsin geophysical modelling. The maximum allowable Courant number is approximately two orders ofmagnitude larger than that of a standard explicit DG formulation;

• several issues concerning the efficient implementation of the method are addressed, with particular at-tention to the derivation of a discrete pseudo–Helmholtz problem for the sole pressure variable by anappropriate static condensation technique, in the same spirit as usually done in finite element hybridizedmethods for the numerical treatment of elliptic problems. Preliminary investigations seem to indicate

Chapter 6. Conclusions and Future Work

that the adoption of the static condensation procedure increases the overall computational efficiency ofthe algorithm by a factor 5 compared to fully explicit DG formulation.

Future extensions and applications of the numerical methodologies proposed in the thesis may include:

• the application of the SLDG technique to large systems of advection diffusion reaction equations usedin atmospheric chemistry modelling, air quality and marine biogeochemistry. Preliminary results inthis direction have already been obtained for passive tracer transport computed on a realistic grid withwind fields obtained with the Lokal–Modell meteorological code. Also, the possibility of exploitingthe methods discussed in this thesis in the framework of the STRATOS Shallow Water Code (describedin [MQS99]) is currently being investigated;

• the application of the FCT monotonization approach in the context of traditional DG formulations tocompute monotone solutions with minimum numerical dissipation;

• the development of a three–dimensional code based on the SIDG formulation which can be interfaced toexisting physical modules to test the effectiveness of the approach in the simulation of realistic test cases.

From the theoretical viewpoint, the following topics emerge from this thesis as possible themes for furtherresearch work:

• the extension of the SLDG formulation, including the FCT monotonization procedure, to the case ofcompressible flows;

• the investigation of optimal strategies to couple advection and diffusion mechanism within the SLDGframework, without resorting to the operator splitting technique;

• the inclusion in the SLDG framework of reaction terms;

• a more complete stability analysis of the SIDG formulation, taking into account the effect of the basic–state atmosphere stratification;

• the use of preconditioning techniques in the solution of the elliptic pseudo–Helmholtz problem obtainedwith the static condensation procedure described in Sect. 4.8 for the SIDG method;

• the use of more sophisticated techniques to treat the open boundary conditions in the SIDG formulationthan the absorbing layer approach adopted in this thesis;

• the inclusion of the Coriolis force terms in the SIDG formulation to tackle large scale phenomena;

• the combination of the SI and SL Discontinuous Galerkin formulations to end up with an unconditionallystable formulation for the solution of the Navier–Stokes Equations, as is usual in the context of finitedifference and continuous finite element formulations.

90

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