Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1026

Stable High-Order FiniteDifference Methods for

Aerodynamics

BY

MAGNUS SVÄRD

ACTA UNIVERSITATIS UPSALIENSISUPPSALA 2004

Dissertation at Uppsala University to be publicly examined in Room 1211, Polacksbacken,Uppsala, Friday, November 12, 2004 at 10:15 for the Degree of Doctor of Philosophy. Theexamination will be conducted in English

AbstractSvärd, M. 2004. Stable High-Order Finite Difference Methods for Aerodynamics (Stabilahögordnings finita differens-metoder för aerodynamik). Acta Universitatis Upsaliensis.Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science andTechnology1026. vii, 25 pp. Uppsala. ISBN 91-554-6063-1

In this thesis, the numerical solution of time-dependent partial differential equations (PDE) isstudied. In particular high-order finite difference methods on Summation-by-parts (SBP) formare analysed and applied to model problems as well as the PDEs governing aerodynamics. TheSBP property together with an implementation of boundary conditions called SAT (SimultaneousApproximation Term), yields stability by energy estimates.

The first derivative SBP operators were originally derived for Cartesian grids. Since aero-dynamic computations are the ultimate goal, the scheme must also be stable on curvilineargrids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBPoperators. Furthermore, aerodynamics often requires addition of artificial dissipation and wederive an SBP version.

With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved.To resolve extremely complex geometries an unstructured discretisation method must be used.Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to beon SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we alsoderive an SBP artificial dissipation for finite volume schemes.

We derive a new set of boundary conditions that leads to an energy estimate for the linearisedthree–dimensional Navier-Stokes equations. The new boundary conditions will be used toconstruct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation,it is necessary to discretise all the second derivatives by using the first derivative approximationtwice. According to previous theory that would imply a degradation of formal accuracy but wepresent a proof that this is not the case.

Keywords:finite difference methods, high-order accuracy, summation-by-parts, stability, energyestimates, finite volume methods

Magnus Svärd, Department of Information Technology, Division of Scientific Computing. UppsalaUniversity. Box 337, SE-751 05 Uppsala, Sweden

c© Magnus Svärd 2004

ISBN 91-554-6063-1ISSN 1104-232Xurn:nbn:se:uu:diva-4621 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-4621 )

to my dear wife Jessica...

List of Papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Svärd, M. (2004) On Coordinate Transformations for Summation-by-Parts operators.Journal of Scientific Computing, 20(1)*

II Mattsson, K., Svärd, M., Nordström, J. (2004) Stable and Ac-curate Artificial Dissipation. Journal of Scientific Computing,21(1)*

III Mattsson, K., Svärd, M., Carpenter, M. H., Nordström, J. (2003)Accuracy Requirements for transient aerodynamics,Paper No.2003-3689, 16th AIAA Computational Fluid Dynamics Confer-ence, Orlando, Florida***

IV Svärd, M., Mattsson, K., Nordström, J., Steady State Computa-tions Using Summation-by-Parts Operators,to appear in Journalof Scientific Computing

V Svärd, M, Nordström, J. (2004) Stability of finite volume approx-imations for the Laplacian operator on quadrilateral and triangu-lar grids,Applied Numerical Mathematics, 51(1)**

VI Gong, J., Svärd, M., Nordström, J. (2004) Artificial Dissipa-tion for Strictly Stable Finite Volume Methods on UnstructuredMeshes. Sixth World Congress on Computational Mechanics inconjunction with Second Asian-Pacific Congress on Computa-tional Mechanics, Beijing, China

VII Nordström, J., Svärd, M., Well-Posed Boundary Conditions forthe Navier-Stokes Equations,Technical Report 2003-052, submit-ted to SIAM Journal on Numerical Analysis

VIII Svärd, M., Nordström, J., (2004) On the Order of Accuracy forDifference Approximations of Inital–Boundary Value Problems,Technical Report 2004-040

*Reprints were made with permission from Kluwer Academic/Plenum Publishers.**Reprints were made with permission from Elsevier.***Reprints were made with permission from American Institute of Aeronautics andAstronautics.

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Contents

1 Introduction . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Stability .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Summation-by-parts Schemes . . . . . . . . . . . . . . . . . . . . . . . . .21.3 Continuous and Discrete Energy Estimates . . .. . . . . . . . . . . . 31.4 The Papers Mutual Dependence . . . . . . . . . . . . . . . . . . . . . . . .41.5 Future Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

2 Summary of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132.6 Paper VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .142.7 Paper VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .172.8 Paper VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18

3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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1 Introduction

This thesis addresses numerical solutions of time-dependent partial differen-tial equations (PDE). In particular we aim to solve the Euler and Navier-Stokesequations but the solution techniques in this thesis may also be used on othertime-dependent problems, such as Maxwell’s equations of electromagnetics.

There is a variety of different solution methods for time-dependent partialdifferential equations. The main classes are spectral methods, finite elements,finite differences and finite volumes. They all have different strengths anddrawbacks, and each have several subgroups.

In general, a numerical method/solver should:• enable efficient computations• have low memory requirements• yield an accurate solution to the PDE.

Finite difference approximations in space utilise the computer resourcesefficiently and hence meet the first requirement. One also has to choose timeintegration scheme with care to obtain an efficient solver. However, that hasnot been the topic of this thesis and for simplicity we use explicit Runge-Kuttamethods in time.

The second requirement can be met by raising the order of the scheme,which decreases the number of grid points (memory) required for the sameaccuracy of the solution. In fact, even the first point gains by raising the ordersince less computations are needed for the same spatial accuracy.

Finally, neither low nor high order is sufficient to get an accurate solutionof the PDE. According to the famous paper by Lax and Richtmyer [14], aconsistent approximation of a well-posed PDE will converge if and only if thescheme is stable.

This gives us the topic of this thesis. Namely, considering well-posednessof the PDE and stability of high-order finite difference schemes.

1.1 StabilityBeginning with the article by Lax and Richtmyer [14] it was proved that forlinear PDEs, stability of a consistent numerical scheme is necessary and suf-ficient for convergence of the numerical solution to the true solution. Thistheorem is of great importance since it is in general easier to prove stability

1

than proving convergence directly.Hence, stability proofs for various PDEs have been the subject of exten-

sive research. For a Cauchy problem, von Neumann analysis gives a neces-sary and often sufficient condition for stability. However, for initial-boundaryvalue problems the von Neumann criterion is still necessary for stability butno longer sufficient. Then the energy method or GKS-analysis [9] can beused to prove stability. These techniques are usually possible to use on modelproblems but for realistic problems they are often too complicated.

A simplification is to consider the semi-discrete problem where time is leftcontinuous. This approach is justified by Kreiss and Wu [13], where theyshow that a stable semidiscretisation can be discretised in time using Runge-Kutta schemes such that the fully discrete problem is stable. The semi-discretecounterpart of the GKS-analysis is obtained by Laplace transforming time andanalysing the spatial operator (see for example the text book [8] and [3, 1]).This is still a very difficult task and virtually impossible for high-order finitedifference methods and realistic problems.

In this thesis, specific discretisation schemes satisfying a summation-by-parts (SBP) rule are used. These are designed to yield energy estimates incombination with specific boundary procedures, even for high-order finite dif-ference schemes. In fact, they can be used to obtain energy estimates even forsystems of PDEs in several space dimensions and complex geometries whichwill be demonstrated in this thesis.

Lax and Richtmyer’s theorem implies that convergence to the true solutionat a fixed timeT is achieved as the grid sizeh → 0. With this definition ofstability the error of the solution may, in fact, grow exponentially in time. Aspointed out by Carpenter et al. in [3], this is not optimal for time-dependentPDEs and they suggest that a scheme should be strictly stable. Strict stabil-ity is a more restrictive definition that does not allow an exponential growthof the error. In Paper IV we arrive at the same conclusion when studyingconvergence to steady state with time marching.

1.2 Summation-by-parts SchemesFinite difference schemes are relatively easy to code and utilise computers ef-ficiently. Most common are low-order methods (first and second order) whichare easy to analyse in terms of stability. Naive ways of implementing bound-ary conditions often result in stable schemes and theoretically it is possibleto analyse the effect on stability from the boundary conditions. However, theaccuracy is often not satisfying.

A high-order finite difference scheme resolves the solution much better butthe analysis of boundary conditions becomes complex and it is not trivial toimplement boundary conditions in a stable manner. Some examples on the

2

use and analysis of high-order finite difference schemes are [1, 2, 6, 3, 7, 26,25, 10, 29, 28].

In the work by Kreiss and Scherer [11, 12], which was followed by Strand[23, 24] high-order finite difference operators with a summation-by-parts (SBP)property were derived for first derivative approximations. Stability for theseschemes can easily be proven with energy estimates for equations in one spacedimension.

Carpenter et al. [4] introduced a new way to implement boundary condi-tions weakly for finite difference methods, with the Simultaneous Approxi-mation Term (SAT) technique. The technique involves penalty terms whichmake energy estimates for PDEs in several space dimensions possible. In aseries of articles by Nordström and Carpenter [5, 18, 19] this technique wasdeveloped for summation-by-parts operators and they also use this techniqueto patch grids together, proving stability and conservation for such interfaces.The patching of different grids allows the finite difference technique to beused for problems with more complex geometries.

In Carpenter, Nordström and Gottlieb [5], a summation-by-parts form forsecond derivative approximations was proposed and the operators derived. Amore optimal version of the second derivatives was later derived in Mattssonand Nordström [16].

Another way to treat boundaries and interfaces was given by Olsson [21,22] where projection operators for implementing boundary conditions wereused. A discussion and numerical experiments comparing the properties ofthe different boundary treatments for SBP schemes were done by Mattsson[15].

In this thesis, the ambition is to use and contribute to the development ofSBP-SAT schemes.

1.3 Continuous and Discrete Energy EstimatesBelow, a brief introduction to SBP-SAT schemes follows and their analogy tothe integration-by-parts property. Consider the equation,

ut +ux = 0, 0≤ t ≤ 1, 0≤ x≤ 1

u(0, t) = g(t), (1.1)

u(x,0) = f (x).

To prove well-posedness,u needs to be bounded. Multiply (1.1) byu andintegrate,

(∫ 1

0u2dx)t +u(1, t)2 = g(t)2 (1.2)

3

Introduce theL2-norm,∫ 1

0 u2dx = ‖u‖2, and solve the ordinary differentialequation (1.2),

‖u‖2 +∫ T

0u(1, t)2dt = ‖ f‖2 +

∫ T

0g(t)2,

i.e. u is bounded. Introduce a grid,xi = h · i, i = 0, ..,N, h = 1/(N − 1).Define the approximate solutionv = (v0(t), ..,vN(t))T on the grid. An SBP-SAT discretisation of (1.1) is,

vt +Dv = σP−1e0(v0(t)−g(t)),

whereσ is a parameter to be determined ande0 = (1,0, ...,0)T . The right-handside is the penalty term defining the SAT.D is a matrix such thatDv≈ vx. IfD is an SBP operator it has the following properties:

D = P−1Q,

P = PT ,

xTPx > 0 for all x,

Q+QT = B = diag(−1,0, ...,0,1).

ThenP is used to define a normvTPv= ‖v‖2P. The analogous derivation of the

continuous energy estimate is,

vTPvt +vTQv = σvTe0(v0−g),(‖v‖2

P)t +vT(Q+QT)v = 2σvTe0(v0−g),(‖v‖2

P)t −v20 +v2

N = 2σ(v20−v0g).

Stability (that is,v bounded withg = 0) is achieved forσ ≤ −1/2. Withσ = −1 we have,

(‖v‖2P)t +v2

N = g2− (v0−g)2.

i.e. the same estimate (with an additional small dissipative term) is obtainedas in the continuous case (1.2).

1.4 The Papers Mutual Dependence

Our main goal is to be able to predict that an approximate solution is theoutcome of a computation. That is done by proving well-posedness of themathematical problem and stability of the numerical scheme. Hence, the pa-pers address a number of problems that will take us a bit further along the pathtowards a reliable and efficient numerical scheme. Each paper has raised new

4

questions which have been investigated in subsequent papers.

Aerodynamic computations are typically performed on curvilinear grids.Paper I studies the effect of curvilinear grids on the stability of SBP schemes.

In Paper II an SBP compatible artificial dissipation is derived. Artificialdissipation is used to kill spurious oscillations that usually appear due to non-linear effects in aerodynamic computations.

A number of computations with the Euler equations comparing high andlow order schemes are performed in Paper III. The test case is a vortex hittinga NACA0012 and we show that high-order methods give much more accurateresults. We also use the artificial dissipation derived in Paper II.

In Paper IV we study steady state solutions around a NACA0012. Efficientcomputations to steady state are needed since steady state solutions serve asinitial data to time-dependent computations such as those in Paper II. Im-proved difference operators are proposed that enhance the convergence speedto steady state. Furthermore, we show that the theoretical conclusions fromthe linear model problem in Paper I are relevant in a nonlinear computation aswell.

So far, mainly hyperbolic PDEs and in particular the Euler equations havebeen studied. We aim for the Navier-Stokes equations and hence we also needto analyse dissipative terms, i.e. second derivatives.

Paper V is a detour to finite volume schemes. The motivation is the pos-sibility of connecting an unstructured method to the finite difference schemeand utilise the strength of an unstructured mesh with respect to resolving verycomplex geometries. Then it is necessary that the finite volume scheme can beinterpreted in the SBP framework. We show that a finite volume approxima-tion of the Laplacian can be viewed as an SBP operator with a new boundarytreatment. We also show that severe restrictions on the grid are needed in orderto get a consistent approximation with the basic scheme. Paper VI is a con-tinuation of Paper V. We use the Laplacian approximation to construct strictlystable first, second and fourth order artificial dissipation operators. Extensivenumerical experiments show their efficiency.

In Paper VII we address the other important issue of this thesis: well-posedness. We derive a novel set of well-posed boundary conditions for theNavier-Stokes equations using energy estimates.

Finally, a study on the accuracy requirements at the boundary for secondderivatives is done in Paper VIII. We show that the boundary scheme canbe approximated with two orders less accuracy than the internal scheme. Thisgives us the possibility to use the first derivative twice for the dissipative termsin the Navier-Stokes equations without degrading the overall accuracy, whichshould simplify stability proofs.

5

1.5 Future ProjectA two-dimensional Euler/Navier-Stokes solver developed by Dr. Mark Car-penter1 was used in this thesis for large scale computations. The next naturalstep to take is to construct a three-dimensional Navier-Stokes solver with allthe new theory (some of it presented in this thesis) implemented. This workhas begun in collaboration with Dr. Mark Carpenter, Prof. Jan Nordström2

and Dr. Nail Yamaleev3. The code is programmed in a modular fashion en-abling several programmers to code independently. Further, it will allow amore efficient parallelisation than the previous 2D-code.

A simple test problem computed with the new code is shown in Figure 1.1.It is a viscous shock computed with a 5th order method. The grid (Figure 1.1)consists of two blocks and the solutions are patched together with the penaltytechnique at the interface. The shock is introduced through the left boundaryand integrated tillT = 1, when the shock is situated at the block interface.Note the smooth solution across the interface in Figure 1.1.

Figure 1.1: Left: Viscous shock atT = 1. Solution field of densityρ . Right: Two-block grid.40×41and41×41points in the respective blocks.

The shock has an analytical solution which allows us to verify the order ofaccuracy of the method. The convergence rate was4.5. The discrepancy (from5) can be explained by the locality of the shock. Depending on where theshock is, the boundary error or internal error will dominate. If the convergencerate is measured atT = 0.75 we obtain order5.4. In both cases we shouldapproach 5 with sufficient grid refinement.

1NASA Langley Research Center2The Swedish Defence Research Agency and Uppsala University3Center for Aerospace Research, North Carolina A&T State University, Greensboro

6

2 Summary of Papers

2.1 Paper I

In [11, 12] several SBP operators for different orders of accuracy were derived.Those can be divided into two categories, diagonal norm schemes and blocknorm schemes. In diagonal norm schemes,P is diagonal and in block normschemes,P has non-zero blocks of fixed sizes at the upper-left and lower-rightcorners. (It is an abuse of notation to callPa norm but we do that for simplicitywhen no confusion can occur.) If block norms are used, a2pth order accurateinterior scheme can be closed with order2p−1 at the boundary, which leadsto order2p globally for hyperbolic problems. However, for diagonal normschemes a2pth order accurate scheme is maximally closed with orderp atthe boundary resulting in(p+ 1)th global order of accuracy for hyperbolicproblems.

Introduce a mappingx(ξ ) between a non-Cartesian gridx and a Cartesianξ (x). Then (1.1) becomes,

vt +∂ξ∂x

vξ = 0.

A discretisation is done in the Cartesianξ -space,

J−1vt +P−1Qv= σP−1e0(v0−g),

whereJ = diag((ξx)0,(ξx)1, ...,(ξx)N), i.e. the mapping evaluated in the gridpoints. The energy method yields,

vTPJ−1vt +vTQv= vTσe0(v0−g).

If PJ−1 is symmetric and positive definite it can be used to define a normand boundedness ofv follows. For diagonalP that is trivially fulfilled but forblock norms it is not. In the paper, it is proven that it is not possible to changeJ for block norm schemes, such thatPJ−1 defines a norm without loosing theorder of accuracy. This implies that there are no energy estimates for stretchedand/or curved grids if block norms are used.

In Figure 2.1 the time evolution of the amplitude of a periodic sine waveis shown. The solution is computed with both a 4th order block and diagonalnorm scheme on a stretched grid. The growth with the block norm scheme is

7

clearly seen.

0 100 200 300 400 500 600 700 8000.95

1

1.05

1.1

1.15

1.2

1.25

t

rela

tive

ampl

itude

0 200 400 600 800 1000 1200 1400 1600 1800 20000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

rela

tive

ampl

itude

Figure 2.1: Amplitude (normalised with initial amplitude) of sine wave on stretchedgrid. Upper: 4th order block norm scheme.Lower: 4th order diagonal norm scheme.

2.2 Paper IIIn most aerodynamic calculations small (or worse large) oscillations appearin the solution due to nonlinear effects. Those are often killed by the additionof artificial diffusion. It is well known how to do this but few have beenconcerned with the stability of the resulting scheme.

The strength of the SBP-SAT schemes is the stability proofs with energy es-timates and the aim of this paper is to construct artificial dissipation operators

8

that do not destroy the energy estimates. Note that we construct linear dis-sipation operators which are intended neither as entropy fixes nor to captureshocks.

Let Dp denote the undividedpth derivative approximation andP = hP. Ar-tificial dissipation is constructed from even order derivatives. We show thatan artificial dissipation of the formP−1A2p with A2p = −DT

pBpDp does notdestroy the energy estimates ifBp is positive semi-definite.

With the derived artificial dissipation operator we can construct schemesfor,

ut +aux = 0, 0≤ x≤ 1, t ≥ 0, a≥ 0

u(0, t) = g(t),u(x,0) = f (x),

as,

vt +aP−1(Q−hA2p)v = σP−1e0(v0−g).

The energy estimate becomes,

‖v‖2t −av2

0 +av2N +ahvTDT

pBpDpv = 2σ(v20−v0g),

which leads to stability withσ ≤−a/2. With a specific scaling ofB we obtaina standard upwind scheme in the interior. (If insteadut −aux = 0 is consideredwe discretise the spatial derivative withP−1(Q+hA2p).)

The remaining issue is accuracy. In the interiorA2p has the same width asthe first derivative approximation for a2pth order scheme ifB is diagonal.Hence, in order not to increase the amount of work in the computation of theartificial dissipation,Bp should be kept diagonal. However, we can allowBp

to have fixed size blocks in the upper-left and lower-right corners.

The theoretical investigations show that ifBp is diagonal and a smooth func-tion, A2p is at mostpth order accurate at the boundary resulting in (p+ 1)thglobal order of accuracy. This is sufficient for diagonal norm schemes but notfor block norm schemes.

To improve the order of accuracy at the boundary, it is shown thatBp cannot be chosen independently ofh or N. Hence, a simple solution is to chooseBp diagonal andO(1) in the interior. At a fixed distance from the boundarywe letBp decrease smoothly toO(hp−1). Then,Ap is order2p in the interiorand order2p−1 near the boundary.

Computations show the efficiency of the new artificial dissipation operatorsfor linear problems and an example where a non-energy stable dissipationintroduces time growth is shown. Further, the nonlinear Burgers’ equationis considered where the artificial dissipation stabilises the solution. As a last

9

test, the nonlinear Euler equations are considered. A vortex with an analyticalsolution is propagated in free stream. The solution was computed with a 3rdand a 5th order upwind scheme. One example is seen in Figure 2.2, wherethe third order upwind scheme is compared with the non-dissipative fourthorder scheme. With dissipation the vortex propagates with the free stream andpasses over the block interface. Without dissipation there are oscillations andthe vortex breaks down before it reaches the block interface.

Finally, the correct order of accuracy is measured with grid refinements forall the different test cases.

0 5 10 15 20 25 30 35 40−10

−5

0

5

10Pressure, 3rd order upwind, t = 20

x

y

0 5 10 15 20 25 30 35 40−10

−5

0

5

10Pressure, 4th order, t = 14

x

y

Figure 2.2: Pressure contour of vortex in free stream.Upper: With dissipation.Lower: Without dissipation. In this case the vortex breaks down a short momentafter T=14.

2.3 Paper III

To test the efficiency of the SBP-SAT on a more realistic problem, we considerthe Euler equations. We use Lax-Friedrichs flux splitting to construct upwindand downwind fluxes which is a convenient way of introducing artificial dis-sipation. For simplicity, consider the one-dimensional problemut + Fx = 0.

10

ThenF is split as follows,

F = Au=A+λ

2u+

A−λ2

u = A+u+A−u

whereλ is the largest eigenvalue ofA. Next, we construct,

D+ = P−1(Q+R), D− = P−1(Q−R)

and obtain a semidiscretisation,

vt +D+A+u+D−A−u = 0.

R is the artificial dissipation constructed in Paper II, that is,R= −Ap.When considering large scale computations, it is necessary to parallelise

the code. The multi-block structure of the scheme offers a simple way of par-allelism by assigning one block to each processor. This is not optimal in thesense that it does not give perfect load balance but the amount of communi-cation is very low. However, with some care when constructing the grid asatisfying load balance can be achieved.

The application we consider is the NACA0012 airfoil in free stream witha vortex convected towards the airfoil and finally interacting with it. Thisis a good example of the flexibility of multi-block grids. Not only need theairfoil be well resolved but if the grid is not sufficiently dense in a narrowregion from the inflow to the airfoil the vortex will dissipate before reachingthe airfoil. This can not be achieved with a standard one-block C-grid.

The solution is computed with a 3rd and a 5th order method. As an exampleof the accuracy obtained with a standard CFD code we also compute the solu-tion with a 2nd order scheme. As expected the accuracy drops with the order,but it is seen in the figures that there is a qualitatively big difference between3rd and 2nd order. This is also seen in the lift and drag coefficients (cl andcd)although the difference is less pronounced on integrated properties.

In Figure 2.3, solutions computed with a 2nd and a 5th order scheme isdisplayed. Note the distorted vortex (large dispersion error) in the 2nd ordersolution.

2.4 Paper IVOften, as in Paper III a steady state solution is needed before time-dependentphenomena can be accurately studied. Also, steady state solutions are im-portant in their own right. Hence, we study convergence to steady state inrealistic computations for the Euler equations. The main scheme of this paperis the diagonal norm scheme with 8th order internal and 4th order boundaryaccuracy, i.e. 5th order global accuracy. A numerical study reveals that a sig-

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Mach number, 2nd order. Fine grid

x

y

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Mach number, 5th order upwind. Fine grid

x

y

Figure 2.3:Mach number contours of vortex at the trailing edge of NACA0012.Up-per: 2nd order scheme.Lower: 5th order scheme.

nificant reduction of the spectrum of the difference operator can be achievedby changing the 3 free parameters of the operator. The spectral radius is re-duced by a factor of 10 and accordingly a 10 times larger time step can be used.This is verified numerically in steady state computations around a NACA0012airfoil.

Further, we study the effect of energy stability with respect to convergenceto steady state. Since a multi-block curvilinear grid is used to resolve theairfoil, a block norm scheme is not energy stable but a diagonal norm schemeis, as discussed in Paper I. In a series of computations we see that the 5th orderdiagonal norm scheme converges with less dissipation than the 5th order blocknorm scheme. In Figure 2.4 an example of the convergence histories of the

12

5th block and diagonal norm schemes with the same amount of dissipation isshown. The block norm scheme fails to converge to steady state. This showsthat energy stability is of importance in realistic computations.

0 2 4 6 8 10 1210

−14

10−12

10−10

10−8

10−6

10−4

10−2

Convergence history, 5:th order case

time

|pn+

1 − p

n|

block normdiagonal norm

Figure 2.4:Convergence history for fifth order block and diagonal norm schemes.

With Lax definition of stability [14], it can be claimed that both schemes arestable, but the block norm scheme has incorrect time behaviour. Hence, wepropose that for steady state computations, strictly stable schemes that ensuresthat the time behaviour is correct, should be used.

At last, we compare the SBP schemes with another high-order non-energystable scheme (see [27]). That scheme is stable on some problems but on ourstandard test problem it is not.

2.5 Paper VOne of the strengths of the SBP-SAT technique for finite difference methods isthe possibility to use multi-block grids. That allows treatment of rather com-plicated geometries. Still, the need to handle very complex geometries and/orsmall geometrical details require unstructured grids. One method that runson unstructured meshes is the node centred finite volume scheme. In [20, 17]Nordström et al. study commonly used finite volume methods and concludethat they can be interpreted in an SBP framework. That gives the possibility ofconnecting a finite volume and the more efficient finite difference scheme in astable manner. Their work is followed up in Paper V where we study a finitevolume approximation of the Laplacian. A second derivative approximationof SBP type was proposed in [5] to have the form,

D2 = P−1(A+DS).

13

P is the same norm as for the first derivative.A is a negative definite matrixand essentially the interior second derivative stencil.D = diag(−1,0, ..,0,1)andSu≈ ux at the boundary.P andA are determined by the scheme we arestudying and we show thatA is negative definite. In addition we derive a formof DS that is accurate if the internal scheme is accurate.

That formulation is used in computations of solutions of the wave equa-tion which reveal some problem with the accuracy of the method. Furtheranalysis shows that the finite volume approximation of the Laplacian is onlyconsistent on equilateral grids (of any shape). (Note that, the inconsistencyappears already in the internal approximation and has nothing to do with thenew boundary treatment.) For grids satisfying that restriction, the approxima-tion including the proposed boundary treatment is second order accurate andstable.

Two examples of computations with the wave equation are shown in Fig-ure 2.5. The solution on the equilateral mesh (upper subfigure) is smoothand fulfills the boundary conditions which are implemented with the proposedboundary procedure. On the other hand, the solution on the triangulated Carte-sian mesh (lower subfigure) clearly has oscillations and it is not zero at theboundary, i.e. it does not fulfill the boundary conditions.

As a comparison the second derivative resulting from double applicationof the first derivative finite volume approximation is analysed. Also, for thisapproximation similar accuracy restrictions must be fulfilled to obtain a con-sistent scheme.

2.6 Paper VI

The Laplacian operator studied in Paper V was shown to be stable for initial-boundary value problems. To obtain satisfying accuracy the grid needs tobe built of equilateral polygons. However, if the Laplacian is scaled as anundivided difference we obtain a second order artificial dissipation with norestrictions on the grid for consistency. The artificial dissipation will not de-stroy the stability of the basic finite volume scheme. Precisely, as the artificialdissipation constructed in Paper II.

Dividing, the second order artificial dissipation with the grid size we obtaina first order dissipation that can be used to dissipate shocks. Further, a doubleapplication of the second order operator yields a fourth order dissipation.

An example of the action of the dissipation is shown in Figure 2.6 for thetwo-dimensional convection equation. The initial data is a sine wave with ran-dom perturbations. As is seen in the figure the dissipation effectively reducesthe high frequency oscillations.

14

00.2

0.40.6

0.81

0

0.5

1−0.05

0

0.05

0.1

0.15

0.2

x

v

y

00.2

0.40.6

0.81

0

0.5

10

0.1

0.2

0.3

0.4

xy

v

Figure 2.5:Numerical solution of the two-dimensional wave equation.Upper: Gridconsisting of equilateral triangles.Lower: Triangulated Cartesian grid.

15

−1−0.8

−0.6−0.4

−0.20

0

0.2

0.4

0.6

0.8

1−1.5

−1

−0.5

0

0.5

1

1.5

x

The numerical results at T=0.5

y

−1−0.8

−0.6−0.4

−0.20

0

0.2

0.4

0.6

0.8

1−1.5

−1

−0.5

0

0.5

1

1.5

x

The numerical results at T=0.5

y

Figure 2.6: Convection equation. Initial data is a sine wave with random perturba-tions. Upper: Solution atT = 0.5 with dissipation.Lower: Solution atT = 0.5without dissipation.

16

2.7 Paper VIISo far, only the numerical stability has been addressed in the articles. A suc-cessful numerical computation relies heavily on well-posedness of the PDE.The main focus is aerodynamics and the governing equations are the Navier-Stokes equations, sometimes simplified to the Euler equations. The questionof well-posedness of the non-linear Navier-Stokes equations is an importantand yet unsolved problem. However, much insight can be gained by studyingthe linearised Navier-Stokes equations. In Paper VII we study the latter setof equations and derive an energy estimate. We present a new way of deriv-ing the number of boundary conditions and we also give a new set of linearlywell-posed boundary conditions. By deriving well-posedness with energy es-timates we have paved the way for a discrete energy estimate for the SBP-SATscheme which would prove stability.

As an example of the technique we use in Paper VII, consider the one-dimensional equation,

ut +A1ux = εA11uxx, 0≤ x≤ 1, t ≥ 0 (2.1)

whereA1 andA11 are symmetric matrices,u the solution vector andε > 0 aconstant. Multiply (2.1) byuT and integrate.∫ 1

0uTut dx+

∫ 1

0uT(A1u− εA11ux)xdx= 0

Introduce the norm‖u‖2 =∫ 1

0 uTudx. We have,

‖u‖2t + ε

∫ 1

0uT

x A11uxdx+wTBw|10 = 0,

wherew = (uT ,A11ux)T and,

B =

(A1 −εI

−εI 0

).

I is the identity matrix and0 denotes the zero matrix. If the eigenvalues andeigenvectors ofB could be derived, it would be possible to splitB into a posi-tive and negative part at each boundary and supply the non-bounded part withboundary conditions. That is what we call the characteristic boundary condi-tions of equation (2.1).

In the Navier-Stokes case one component ofw is zero. In our approach weremove the corresponding row and column ofB by constructing aB. Then wesplit B in a positive and a negative part. This is achieved through a rotation ofB to block-diagonal form and then calculating the eigenvalues of the resultingmatrix. We obtain both the number of boundary conditions and a novel set of

17

boundary conditions to specify at each boundary.

2.8 Paper VIIIThe Navier-Stokes equations include dissipative terms. Hence, we need tostudy the behaviour of second derivative approximations. One issue that arisesis what order of accuracy one needs in the boundary closure. Consider,

ut = uxx, 0≤ x≤ 1, t ≥ 0,

L0,1u = g0,1(t), (2.2)

u(x,0) = f (x),

where f ,g0,g1 are initial and boundary data. Equation (2.2) is discretised by,

vt = D2v+B(g0,g1). (2.3)

D2 is an approximation of the second derivative operator andB holds theboundary data. Ifu is inserted into (2.3) (interpreted as a grid function) andsubtracted from (2.3) we obtain,

et = D2e+T,

wheree = u− v andT is the truncation error. For high-order methods theboundary schemes may become complicated and if the order of the boundaryschemes is lowered, it might be easier to prove stability. In general we have,

T = (O(hr), ..,O(hr),O(hp), ...,O(hp),O(hr), ..,O(hr)).

What is the minimumr that results in global order of accuracyp?Numerical evidence and previous investigations indicate that the accuracy

can be degraded two orders at the boundary, i.e.r = p− 2. Paper VIII ad-dresses that question and by demanding pointwise boundedness of the numer-ical solution, two orders less accuracy is allowed. Further, it is shown thatSBP schemes with a discrete energy estimate are pointwise stable. The theo-retical derivations even suggest a more general conclusion. With a pointwisestable scheme, one can degrade the order at the boundary as many orders asthe order of the principal part of the PDE.

18

3 Acknowledgements

I would like to express my sincere gratidude to my advisor Jan Nordström forhis guidiance and expertise, and also for studying my manuscripts in minutedetail. Not only have you taught me how to conduct research, but also im-proved my physical shape during our Wednesday runs.

Ken Mattsson, thank you for being co-author on three of the papers and forpushing me to new records in the running tracks, while chasing your back farahead. Mark Carpenter, thank you for being co-author and for keeping me ontrack with a clear goal during our endeavour towards a new code. Thank you,Jing Gong for being co-author. It has been good to work with you.

I would also like to thank, Bertil Gustafsson and Per Lötstedt for goodcourses and for always being ready to discuss my research.

Thank you, Mum and Dad for your support and for showing interest mywork. Last but not the least, thank you Jessica, for your love and patienceduring these years.

This work has partially been funded by The Swedish Defence ResearchAgency (FOI) and the former Aeronautical Research Institute of Sweden (FFA).

19

4 Summary in Swedish

Stabila högordnings finita differens-metoder föraerodynamikDenna avhandling handlar om numerisk lösning av tidsberoende partiella dif-ferentialekvationer (PDE), speciellt sådana som styr aerodynamik. Detta görsmed högordnings finita differens-metoder vilka effektivt utnyttjar datorresurs-erna. Traditionellt har det varit svårt att konstruera en stabil randbehandlingmen för en viss klass av metoder som uppfyller en partiell summationsregel(jmfr partiell integration) är det möjligt att visa stabilitet för många PDE.Dessa (SBP-) metoder har studerats i denna avhandling.

En aerodynamisk beräkning kräver i allmänhet artificiell dissipation föratt dämpa oscillationer i lösningen. Vi har härlett en artificiell dissipationsom uppfyller samma partiella summationsregel som approximationen själv.Således bevaras energiuppskattningen och därmed stabiliteten. Vidare har ap-proximationer på kroklinjiga koordinatsystem studerats. För att bevara energi-uppskattning och stabilitet i detta fall kan man bara använda en delmängd avSBP-metoderna.

Vi har gjort realistiska aerodynamiska beräkningar med Eulers ekvationerför en vingprofil som interagerar med en virvel. I dessa beräkningar användsden nya artificiella dissipationen. Vi visar att högordningsmetoder ger betyd-ligt noggrannare resultat än lågordningsmetoder.

Eftersom vi studerar flödet runt en vingprofil så är beräkningsnätet kroklin-jigt. Vi påvisar att den del av SBP-metoderna som bevarar energiuppskatt-ningar för kroklinjiga nät kräver mindre dissipation för att konvergera till denstationära lösningen.

Som nämnts ovan, så bevisas stabilitet för SBP-metoder via energiuppskatt-ningar. Detta är inte möjligt om inte PDEn själv upfyller en energiuppskatt-ning. Därför har vi härlett en energiuppskattning och rättställda randvillkorför Navier-Stokes ekvationer. Dessa kommer att vara till hjälp då vi ska kon-struera en stabil randapproximation.

Ett inledande arbete inför en eventuell ihopkoppling av finita differens-metoder och en ostrukturerad finit volym-metod har genomförts, där partiellsummationsegenskap hos den senare metoden har studerats. En sådan ihop-koppling skulle göra det möjligt att approximera flöden runt kompliceradegeometrier.

21

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Acta Universitatis UpsaliensisComprehensive Summaries of Uppsala Dissertations

from the Faculty of Science and TechnologyEditor: The Dean of the Faculty of Science and Technology

Distribution:Uppsala University Library

Box 510, SE-751 20 Uppsala, Swedenwww.uu.se, acta@ub.uu.se

ISSN 1104-232XISBN 91-554-6063-1

A doctoral dissertation from the Faculty of Science and Technology, UppsalaUniversity, is usually a summary of a number of papers. A few copies of thecomplete dissertation are kept at major Swedish research libraries, while thesummary alone is distributed internationally through the series ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science and Technology.(Prior to October, 1993, the series was published under the title “ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Science”.)