anadaptivemovingmeshdiscontinuous galerkin method for...

Commun. Comput. Phys.doi: 10.4208/cicp.OA-2018-0317

Vol. 27, No. 4, pp. 1140-1173April 2020

An Adaptive Moving Mesh Discontinuous Galerkin

Method for the Radiative Transfer Equation

Min Zhang1, Juan Cheng2,3, Weizhang Huang4 and Jianxian Qiu5,∗

1 School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005,China.2 Laboratory of Computational Physics, Institute of Applied Physics andComputational Mathematics, Beijing 100088, China.3 Center for Applied Physics and Technology, Peking University, Beijing 100871,China.4 Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, USA.5 School of Mathematical Sciences and Fujian Provincial Key Laboratory ofMathematical Modeling and High-Performance Scientific Computing, XiamenUniversity, Xiamen, Fujian 361005, China.

Received 7 December 2018; Accepted (in revised version) 4 May 2019

Abstract. The radiative transfer equation models the interaction of radiation with scat-tering and absorbing media and has important applications in various fields in scienceand engineering. It is an integro-differential equation involving time, frequency, spaceand angular variables and contains an integral term in angular directions while be-ing hyperbolic in space. The challenges for its numerical solution include the needsto handle with its high dimensionality, the presence of the integral term, and the de-velopment of discontinuities and sharp layers in its solution along spatial directions.Its numerical solution is studied in this paper using an adaptive moving mesh discon-tinuous Galerkin method for spatial discretization together with the discrete ordinatemethod for angular discretization. The former employs a dynamic mesh adaptationstrategy based on moving mesh partial differential equations to improve computa-tional accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency aredemonstrated in a selection of one- and two-dimensional numerical examples.

AMS subject classifications: 65M50, 65M60, 65M70, 65R05, 65.75

Key words: Adaptive moving mesh, discontinuous Galerkin method, radiative transfer equation,high order accuracy, high resolution.

∗Corresponding author. Email addresses: [email protected] (M. Zhang),cheng [email protected] (J. Cheng), [email protected] (W. Huang), [email protected] (J. Qiu)

http://www.global-sci.com/cicp 1140 c©2020 Global-Science Press

M. Zhang et al. / Commun. Comput. Phys., 27 (2020), pp. 1140-1173 1141

1 Introduction

The radiative transfer equation (RTE) models the interaction of radiation with scatteringand absorbing media, which has important applications in fields such as astrophysics,high energy density physics, nuclear physics, inertial confinement fusion, heat transfer,stellar atmospheres, optical molecular imaging, infrared and visible light in space and theatmosphere, and biomedicine. The RTE is an integro-differential equation with seven in-dependent variables (i.e., time, frequency, space, and angles) for a time-dependent three-spatial-dimensional problem. Containing an integral term and with its high dimension-ality, the RTE presents a challenge in the development of efficient numerical algorithms.On the other hand, the efficient solution of the RTE plays an important role in the studyof radiation hydrodynamics where the RTE is often coupled with the Euler equations, theenergy equation, and the equation of state.

In the past, a number of methods have been developed for the numerical solution ofthe RTE. Those methods can be divided roughly into two categories: stochastic and de-terministic approaches. The Monte Carlo method is a widely used method in the formercategory [24, 31]. On the other hand, deterministic approaches involve discrete approxi-mations of the variables in the RTE. In particular, the discretization needs to be appliedto all coordinates in space and angles. For angular coordinates, the PN method, first in-troduced in [25] and also known as the spherical harmonics method, uses an orthogonal,harmonic basis to approximate the solution. Another approach called the discrete or-dinate method (DOM) [6, 26] employs spectral collocation and the Legendre-Chebyshevquadrature to discretize the integro-differential equation in angular coordinates. DOM iswidely used for the numerical solution of the transport equation [28, 35] due to its highaccuracy, flexibility, and relatively low computational cost. The angle-discretized RTEforms a system of linear hyperbolic equations with a numerical integral term, which canbe discretized in space using a standard method such as a finite difference, finite volume,or finite element method. The discontinuous Galerkin (DG) method is employed for thispurpose in the current work.

The DG method is known to be a particularly powerful numerical tool for the simula-tion of hyperbolic transport problems. It was first used for the RTE by Reed and Hill [36]and theoretically studied by Lesaint and Raviart [27]. The method was later extendedto nonlinear conservation laws by Cockburn and Shu [7–10]. The DG method has theadvantages of high-order accuracy, geometric flexibility, suitability for handling h- andp-adaptivity, extremely local data structure, high parallel efficiency, and a good theo-retical foundation for stability and error estimates. Over the last few decades, the DGmethod has been used widely in scientific and engineering computation.

The objective of the current work is to study an adaptive moving mesh DG method(for spatial discretization) combined with DOM (for angular discretization) for the nu-merical solution of the RTE. Due to its hyperbolic nature, the solution of the RTE candevelop discontinuities or sharp layers along spatial directions, which makes mesh adap-tation an indispensable tool for use in improving computational accuracy and efficiency.

1142 M. Zhang et al. / Commun. Comput. Phys., 27 (2020), pp. 1140-1173

Mesh adaptation methods can be classified roughly into three groups. The first one is h-methods, which generate a new mesh by adding or removing points to an existing mesh.Typically, mesh points are added in regions where the solution variation or error is large,and mesh points are removed in regions where the solution is smooth. The second groupis p-methods where the order of polynomial approximation varies from place to place ac-cording to a certain error estimate or indicator. The third group is r-methods, also calledmoving mesh methods, which relocate mesh point positions while maintaining the totalnumber of mesh points and the mesh connectivity.

Several works have been done in using mesh adaptation for the numerical solutionof the RTE. In [22], an adaptive mesh refinement (AMR) algorithm has been formulatedand implemented for the RTE by minimizing the spatial discretization error. In [13], hp-adaptive DG methods have been developed for the numerical solution of simplified PN

approximations of radiative transfer in non-grey semitransparent media and it has beenfound that it is possible to approximate the radiative field with a significantly lower com-putational cost than solving the equations using the conventional finite element method.In [12], hp-adaptive methods have been developed based on a short-characteristics ap-proach embedded in the discontinuous finite element framework. No work has beendone so far in using moving mesh methods for the numerical solution of the RTE.

In this paper, we study a moving mesh method based on moving mesh partial dif-ferential equations (MMPDEs) [18–20] to solve the radiative transfer equation. An MM-PDE moves the mesh continuously in time and orderly in space and is formulated as thegradient flow equation of a meshing functional. We use a newly developed discrete ap-proach [16] that makes the implementation of the MMPDE method not only significantlysimpler but also much more reliable in the sense that there is a theoretical guarantee formesh nonsingularity at semi and fully discrete levels [17]. The MMPDE method deter-mines the mesh adaptivity for the size, shape, and orientation of mesh elements usinga metric tensor (also called a monitor function) which is matrix-valued function definedthroughout the physical domain.

The full discretization of the RTE includes the discretization in angular directions us-ing DOM, in space using an adaptive moving mesh DG method [29], and in time usingthe backward Euler scheme. The DOM discretization leads to radiative intensity func-tions for different angular directions which can have discontinuities or sharp layers atdifferent location in space. To take this into account, we compute a metric tensor basedon the Hessian for each of these functions and then combine them via matrix intersection(see its definition in Section 4) into a single metric tensor to be used with the MMPDEmethod. Numerical results in Section 5 show that the adaptive moving mesh DG methodwith this strategy works well for problems with single or multiple sharp layers and dis-continuities in the sense that it is able to automatically concentrate the mesh points inthe regions of discontinuities or steep transition layers and is more efficient than its fixedmesh counterpart.

An outline of the paper is as follows. In Section 2, the RTE and DOM are described.The adaptive moving mesh DG method for solving the two-dimensional RTE is presented


in Section 3. The generation of adaptive moving meshes using a new implementation ofthe MMPDE method is discussed in Section 4. A selection of one- and two-dimensionalexamples are presented in Section 5 to demonstrate the mesh adaptation ability of theadaptive moving mesh DG method and its accuracy and efficiency. Finally, Section 6contains conclusions and further comments.

2 The radiative transfer equation

The RTE is an integro-differential equation modeling the conservation of photons [32].We consider a case with one-group, isotropically scattering radiative transfer. The gov-erning equation for this case reads as

1

c

∂I(r,Ω,t)

∂t+Ω·∇I(r,Ω,t)+σt I(r,Ω,t)=

σs

4π

∫

SI(r,Ω,t)dΩ+q(r,Ω,t), (2.1)

where c is the speed of photons, r is the spatial variable, ∇ is the gradient operator withrespect to r, Ω is the unit angular variable, S is the unit sphere, t is time, I(r,Ω,t) is theradiative intensity in the direction Ω, σs ≥ 0 is the scattering coefficient of the medium,σt ≥σs is the extinction coefficient of the medium due to both absorption and scattering,and q(r,Ω,t) is a given source term. The vector r is described by the Cartesian coordinatesx,y,z while Ω is usually described by a polar angle β measured with respect to a fixedaxis in space (such as the z axis) and a corresponding azimuthal angle ϕ. Letting µ=cosβ,then

dr=dxdydz, dΩ=sinβdβdϕ=−dµdϕ.

In this work we consider the numerical solution of (2.1) in one and two spatial dimen-sions. Since the numerical algorithm is similar in one and two dimensions, we describeit only in two dimensions. The equation (2.1) reads in two dimensions as

1

c

∂I(x,y,Ω,t)

∂t+Ω·∇I(x,y,Ω,t)+σt I(x,y,Ω,t)

=σs

4π

∫

SI(x,y,Ω,t)dΩ+q(x,y,Ω,t), (x,y)∈D, (2.2)

where Ω=(ζ,η) and

ζ=sinβcos ϕ=√

1−µ2cosϕ∈ (−1,1), η=sinβsin ϕ=√

1−µ2sinϕ∈ (−1,1).

Denote by n(x,y) the unit outward normal vector of the domain boundary ∂D at thepoint (x,y) and define ∂Din =(x,y)∈∂D |n(x,y)·Ω<0. Then, the boundary conditioncan be expressed as

I(x,y,Ω,t)= g(x,y,Ω,t), (x,y)∈∂Din , (2.3)

and the initial condition is

I(x,y,Ω,0)= I0(x,y,Ω), (x,y)∈D, (2.4)


where g(x,y,Ω,t) and I0(x,y,Ω) are given functions. Note that no boundary condition isneeded in Ω directions.

The RTE (2.2) needs to be discretized in angular, spatial, and time variables. A chal-lenge for solving (2.2) is due to its high dimensionality: it has two dimensions totally inboth angular and spatial coordinates for a one-spatial-dimensional problem and four di-mensions for a two-spatial-dimensional problem. To tackle this challenge, a commonstrategy is to use a high accuracy discretization in angular coordinates. We use thediscrete-ordinate method [6] for this purpose. DOM will be described later in this sec-tion. Another challenge for solving (2.2) is due to the fact that it is hyperbolic in space.This means that its solution can develop discontinuities and sharp layers across the phys-ical domain, which requires a spatial discretization that can handle those structures andmesh adaptation that can provide high resolution in regions around them. An adaptivemoving mesh DG method to be presented in the next section will be used to tackle thischallenge.

DOM [6] is a spectral collocation-type method [5]. Indeed, (2.2) is collocated for afinite number of angular directions while the integral in the angular variable is approxi-mated by the Legendre-Chebyshev quadrature [26]. The collocation points in ζ and η aregiven by

ζm =√

1−µ2i cos ϕj, ηm =

√

1−µ2i sinϕj, m=(i−1)Nc+ j,

where µi, i=1,··· ,Nl denote the roots of the Legendre polynomial of degree Nl and ϕj =(2j−1)/Nc, j=1,··· ,Nc are the nodes based on a Chebyshev polynomial. Specifically, thediscrete-ordinate approximation of (2.2) is given by

1

c

∂Im(x,y,t)

∂t+Ωm ·∇Im(x,y,t)+σt Im(x,y,t)

=σs

Na

∑m′=1

wm′ Im′(x,y,t)+qm(x,y,t), m=1,··· ,Na, (2.5)

where Ωm = (ζm,ηm), m = 1,··· ,Na ≡ Nl Nc, are the discrete angular directions, Im(x,y,t)

is an approximation of I(x,y,Ωm,t), qm(x,y,t) = q(x,y,Ωm,t), and ∑Nam=1wm Im(x,y,t) is a

Legendre-Chebyshev quadrature rule with weights wm>0 for (1/4π)∫

S I(x,y,Ω,t)dΩ.

3 An adaptive moving mesh DG method for the

two-dimensional DOM RTEs

We notice that (2.5) is a system of hyperbolic equations and its solution can developdiscontinuities and sharp layers. The DG method has been known to be a powerfulnumerical tool for the simulation of hyperbolic problems with discontinuous solutions[27, 36]. Mesh adaptation is also crucial to provide accurate resolution of discontinuitiesand sharp layers in the solution and improve computational efficiency. We describe a


DG method for (2.5) on a general adaptive moving mesh in this section and the adaptivemesh movement in the next section.

Specifically, we consider time instants

t0=0< t1< ···< tn < tn+1< ··· .

For the moment, we assume that a triangular mesh, which consists of non-overlappingtriangles covering D completely and whose vertices depend on t, is known at the timeinstants, i.e., T n

h , n = 0,1,··· , are given. We also assume that the mesh keeps the sameconnectivity and the same number of elements and vertices for the whole time period.(The position of the vertices is the only thing that changes with time.) The generation ofsuch a moving mesh is discussed in Section 4.

For t∈ [tn ,tn+1], the coordinates and velocities of the vertices of the mesh are definedas

xj(t)= xnj

tn+1−t

∆tn+xn+1

j

t−tn

∆tn, xj(t)=

xn+1j −xn

j

∆tn, j=1,··· ,Nv,

yj(t)=ynj

tn+1−t

∆tn+yn+1

j

t−tn

∆tn, yj(t)=

yn+1j −yn

j

∆tn, j=1,··· ,Nv,

(3.1)

where Nv is the number of the vertices and ∆tn = tn+1−tn. The corresponding mesh isdenoted by Th(t).

We now describe the DG discretization of (2.5) on Th(t). For any element K∈Th(t),denote its vertices by (xK

1 ,yK1 ), (xK

2 ,yK2 ), (xK

3 ,yK3 ) and its area by |K|. Consider a set of local

orthogonal polynomials of up to degree k in K,

Pk(K,t)=spanφ(K)0 (x,y,t), φ

(K)1 (x,y,t), ··· , φ

(K)L−1(x,y,t), (3.2)

where L=(k+1)(k+2)/2 is the dimension of Pk(K,t) and φ(K)p , p=0,1,··· ,L−1 are the

orthogonal basis functions over the element K. The first three of the basis functions readas

φ(K)0 (x,y,t)=1,

φ(K)1 (x,y,t)=

√2(

x− 1

3

)

,

φ(K)2 (x,y,t)=

√2

2

(

x− 1

3

)

+√

2(

y− 1

3

)

,

where[

xy

]

=1

2|K|

[

(yK3 −yK

1 ) −(xK3 −xK

1 )−(yK

2 −yK1 ) (xK

2 −xK1 )

][

x−xK1

y−yK1

]

.

Then the associated DG finite element space can be defined as

Vkh (t)=Ih

m(x,y,t)∈L2(D) : Ihm(x,y,t)|K ∈Pk(K,t), ∀K∈Th(t). (3.3)


Notice that any function in this space can be expressed as

Ihm(x,y,t)=

L−1

∑p=0

I[p]m,K(t)φ

(K)p (x,y,t), (x,y)∈K, (3.4)

where I[p]m,K, p=0,··· ,L−1, are the degrees of freedom. Moreover, its time derivative can

be written as

∂Ihm(x,y,t)

∂t=

L−1

∑p=0

(dI[p]m,K(t)

dtφ(K)p (x,y,t)+ I

[p]m,K(t)

∂φ(K)p (x,y,t)

∂t

)

. (3.5)

It is not difficult (e.g., see [23]) to show that

∂φ(K)p (x,y,t)

∂t=−Π1(x,y,t)·∇φ

(K)p (x,y,t), (3.6)

where Π1(x,y,t)=(X,Y) is the piecewise linear interpolation of the nodal mesh velocities,i.e.,

X=1

3(xK

1 + xK2 + xK

3 )φ(K)0 −

√2

4(xK

1 −2xK2 + xK

3 )φ(K)1 −

√2

2(xK

1 − xK3 )φ

(K)2 ,

Y=1

3(yK

1 + yK2 + yK

3 )φ(K)0 −

√2

4(yK

1 −2yK2 + yK

3 )φ(K)1 −

√2

2(yK

1 − yK3 )φ

(K)2 .

(3.7)

Combining (3.5) and (3.6), we get

∂Ihm(x,y,t)

∂t=

L−1

∑p=0

dI[p]m,K(t)

dtφ(K)p (x,y,t)−Π1(x,y,t)·∇Ih

m(x,y,t), (x,y)∈K. (3.8)

Multiplying (2.5) by a test function φ(x,y,t) ∈ Vkh (t), integrating the resulting equation

over K, replacing Im(x,y,t) with its approximation Ihm(x,y,t), and using (3.8), we have

∫

K

(1

c

L−1

∑p=0

dI[p]m,K(t)

dtφ(K)p

)

φdxdy+∫

Kσt Ih

m(x,y,t)φdxdy

+∫

K(Ωm−

1

cΠ1(x,y,t))·∇Ih

m(x,y,t)φdxdy

=∫

KσsΨK(x,y,t)φdxdy+

∫

Kqm(x,y,t)φdxdy, (3.9)

where

ΨK(x,y,t)=Na

∑m=1

wm Ihm,K(x,y,t)


and Ihm,K(x,y,t) denotes the restriction of Ih

m(x,y,t) on K. Applying the divergence theo-rem on the third term, we get

∫

K

(1

c

L−1

∑p=0

dI[p]m,K(t)

dtφ(K)p

)

φdxdy+∫

Kσt Ih

m(x,y,t)φdxdy

−∫

K

(

∇·(

φ(Ωm−1

cΠ1(x,y,t))

)

)

Ihm(x,y,t))dxdy

+∫

∂KnK ·(

Ωm−1

cΠ1(x,y,t)

)

Ihm(x,y,t)φds

=∫


∫

Kqm(x,y,t)φdxdy, (3.10)

where nK is the outward unit normal to the boundary ∂K. In the above equation, Ihm(x,y,t)

is discontinuous across the cell boundaries in general and its value thereon is not welldefined. To specify the value, we define the outflow boundary ∂Km+ and the inflowboundary ∂Km− of the element K by

∂Km+=

(x,y)∈∂K∣

∣

(

Ωm−1

cΠ1(x,y,t)

)

·nK(x,y,t)≥0

,

∂Km−=

(x,y)∈∂K∣

∣

(

Ωm−1

cΠ1(x,y,t)

)

·nK(x,y,t)<0

.

(3.11)

It is useful to point out that, in practice, (1/c)Π1(x,y,t) is much smaller than Ωm for mostsituations and ∂Km+ and ∂Km− can be computed using the simpler formulas

∂Km+=(x,y)∈∂K |Ωm ·nK(x,y,t)≥0,

∂Km−=(x,y)∈∂K |Ωm ·nK(x,y,t)<0.(3.12)

Since each interior edge is shared by two triangular elements, the value of Ihm(x,y,t) on

any edge of K can be defined based on its value on K or on the other element sharing thecommon edge with K. These values are denoted by Ih

m(int(K),t)) and Ihm(ext(K),t), re-

spectively. For the upwind numerical flux, we use Ihm(int(K),t) for the outflow boundary

and Ihm(ext(K),t) for the inflow boundary. Thus, we can rewrite (3.10) into

∫

K

(1

c

L−1

∑p=0

dI[p]m,K(t)

dtφ(K)p

)

φdxdy+∫

Kσt Ih

m(x,y,t)φdxdy

−∫

K

(

∇·(

φ(Ωm−1

cΠ1(x,y,t))

)

)

Ihm(x,y,t)dxdy

+∫

∂Km−nK ·(

Ωm−1

cΠ1(x,y,t)

)

Ihm(ext(K),t)φds

+∫

∂Km+nK ·(

Ωm−1

cΠ1(x,y,t)

)

Ihm(int(K),t)φds

=∫


∫

Kqm(x,y,t)φdxdy. (3.13)


Explicit time stepping can cause extremely small time steps due to the high photonspeed. To avoid this difficulty, we use the backward Euler scheme for (3.13), i.e.,

∫

K

(1

c

L−1

∑p=0

I[p]m,K(tn+1)− I

[p]m,K(tn)

∆tnφ(K)p

)

φdxdy+∫

Kσt Ih

m(x,y,tn+1)φdxdy

−∫

K

(

∇·(

φ(Ωm−1

cΠ1(x,y,tn+1))

)

)

Ihm(x,y,tn+1)dxdy

+∫

∂Km−nK ·(

Ωm−1

cΠ1(x,y,tn+1)

)

Ihm(ext(K),tn+1)φds

+∫

∂Km+nK ·(

Ωm−1

cΠ1(x,y,tn+1)

)

Ihm(int(K),tn+1)φds

=∫

KσsΨK(x,y,tn+1)φdxdy+

∫

Kqm(x,y,tn+1)φdxdy,

∀φ∈Vkh (tn+1), K∈Th(tn+1), m=1,··· ,Na. (3.14)

The above equations form a coupled system for the unknown functions Ihm(x,y,tn+1),

m=1,··· ,Na since the function ΨK(x,y,tn+1) contains all of them. To decouple these func-tions from the equations, a functional-type iteration called the source iteration (SI) [28,39](also referred to as the grid sweeping algorithm) has been widely used for solving the sys-tem in a Gauss-Seidel-like manner. To be specific, assuming that the ℓ-th iteration solu-

tions Ih(ℓ)m,K (x,y,tn+1) (for m=1,··· ,Na and K∈Th(tn+1)) are known, we compute the new

approximations Ih(ℓ+1)m,K (x,y,tn+1) element by element in a sweeping direction [11] and

through all angular directions m=1,··· ,Na for each given element. Thus, for K∈Th(tn+1),we have

∫

K

(1

c

L−1

∑p=0

I[p](ℓ+1)m,K (tn+1)− I

[p]m,K(tn)

∆tnφ(K)p

)

φdxdy+∫

Kσt I

h(ℓ+1)m,K (x,y,tn+1)φdxdy

−∫

K

(

∇·(

φ(Ωm−1

cΠ1(x,y,tn+1))

)

)

Ih(ℓ+1)m,K (x,y,tn+1)dxdy

+∫

∂Km−nK ·(

Ωm−1

cΠ1(x,y,tn+1)

)

Ih(ℓ+1)m,K (ext(K),tn+1)φds

+∫

∂Km+nK ·(

Ωm−1

cΠ1(x,y,tn+1)

)

Ih(ℓ+1)m,K (int(K),tn+1)φds

=∫

KσsΨ∗

K(x,y,tn+1)φdxdy+∫

Kqm(x,y,tn+1)φdxdy,

∀φ∈Vkh (tn+1), m=1,··· ,Na, (3.15)


where

Ψ∗K(x,y,tn+1)=

Na

∑m=1

wm I∗m,K(x,y,tn+1),

I∗m,K(x,y,tn+1)=

Ih(ℓ+1)m,K (x,y,tn+1), when available,

Ih(ℓ)m,K (x,y,tn+1), otherwise.

The iteration is stopped when the difference between two consecutive iterates is smaller

than a given tolerance. In our computation, we use maxm‖Ih(ℓ+1)m − I

h(ℓ)m ‖∞ ≤ 10−12. The

source iteration is very effective, taking only a few iterations to achieve convergence formost of the problems tested.

The time integration alternates between solving the physical equation and generatingthe mesh. Starting with the current mesh T n

h and a solution Ihm(x,y,tn), m= 1,··· ,Na, a

new mesh Tn+1

h is generated using the MMPDE moving mesh strategy to be described inthe next section. Then, the DOM-DG scheme (3.15) at tn+1 is solved for the new solutionapproximation Ih

m(x,y,tn+1), m=1,··· ,Na.

4 The MMPDE moving mesh method on triangular meshes

In this section we describe the generation of Tn+1

h based on T nh and numerical solution

Ihm(x,y,tn), m = 1,··· ,Na using the MMPDE moving mesh method [18–20]. The method

utilizes a metric tensor (or called a monitor function) to provide the information of thesize, shape, orientation of mesh elements throughout the domain that is needed for meshadaptation. We use here a new implementation of the method proposed in [16]. A uniquefeature for the numerical solution of the RTE is that the functions Ih

m(x,y,tn), m=1,··· ,Na

which correspond to the radiative intensity at angular directions Ωm, m= 1,··· ,Na mayhave discontinuities and sharp layers at different locations in space. To take this intoaccount, we first compute the metric tensor for each function and then combine all of themetric tensors into a single one.

We start with noting that Tn+1

h and T nh have the same number of the elements (N),

the same number of the vertices (Nv), and the same connectivity. They differ only in thelocation of the vertices. We assume that a reference computational mesh Tc=(ξ1

j , ξ2j ), j=

1,··· ,Nv, which also has the same connectivity and the same numbers of vertices andelements as T n

h , has been chosen. In our computation, we take it as a uniform mesh

(in the Euclidean metric) defined on domain D. Tc stays fixed in the computation. Thegeneration of T

n+1h is through a computational mesh Tc = (ξ1

j ,ξ2j ), j= 1,··· ,Nv which

serves as an intermediate variable.

A key idea of the MMPDE moving mesh method is to view any nonuniform meshas a uniform one in some metric M [15, 18]. The metric tensor M =M(x,y,t) is a sym-metric and uniformly positive definite matrix-valued function defined on D. It provides


the magnitude and direction information needed for determining the size, shape, andorientation of the mesh elements throughout the domain. Various metric tensors havebeen proposed; e.g., see [18, 21]. We here use a metric tensor based on the Hessian of thecomputed solution. Let HK(Ih

m(tn)) be the Hessian or a recovered Hessian of Ihm(tn) on K.

Let the eigen-decomposition of HK(Ihm(tn)) be

HK(Ihm(tn))=Qdiag(λ1,λ2)Q

T.

Denote|HK(Ih

m(tn))|=Qdiag(|λ1|,|λ2|)QT.

The metric tensor is then defined as

MK,m=det(

I+1

αh,m|HK(Ih

m(tn))|)− 1

6(

I+1

αh,m|HK(Ih

m(tn))|)

, ∀K∈Th, (4.1)

where Th denotes a general physical mesh, I is the identity matrix, det(·) is the deter-minant of a matrix, and αh,m is a regularization parameter defined through the algebraicequation

∑K∈Th

|K|det(MK,m)12 =2 ∑

K∈Th

|K|det(|HK(Ihm(tn))|)

13 .

The metric tensor (4.1) is known [21] to be optimal for the L2-norm of linear interpolationerror.

Notice that for each m (1≤m≤Na), MK,m provides the mesh adaptation informationonly for function Ih

m(x,y,tn). To account for all of the functions in the mesh adaptation,we need to combine the metric tensors into a single one. We define

MK =MK,1∩MK,2∩···∩MK,Na ,

where “∩” stands for the intersection of symmetric and positive definite matrices whichis defined as follows. Let A and B be two symmetric and positive definite matrices. Thereexists a nonsingular matrix P such that PAPT=I and PBPT=diag(b1,b2). The intersectionof A and B is then defined as A∩B=P−1diag(max(1,b1),max(1,b2))P−T. Define

DC =(x,y) : (x,y)C(x,y)T<1

for any symmetric and positive definite matrix C. It is not difficult to show that DA∩B ⊆DA∩DB, which gives the meaning of the name “intersection”. Notice that the definitionis not optimal in the sense that DA∩B is not necessarily the biggest ellipse inscribed in DA

and DB.It is known [15, 18] that if Th is uniform in the metric M in reference to the computa-

tional mesh Tc, it satisfies

|K|√

det(MK)=σh|Kc||Dc|

, ∀K∈Th, (4.2)

1

2tr(

(F′K)

−1M−1K (F′

K)−T)

=det(

(F′K)

−1M−1K (F′

K)−T)

12 , ∀K∈Th, (4.3)


where F′K is the Jacobian matrix of the affine mapping: FK :Kc∈Tc→K, MK is the average

of M over K, tr(·) denotes the trace of a matrix, and

|Dc|= ∑Kc∈Tc

|Kc|, σh = ∑K∈Th

|K|det(MK)12 .

The condition (4.2), called the equidistribution condition, determines the size of elementsthrough the metric tensor M. On the other hand, (4.3), referred to as the alignment con-dition, determines the shape and orientation of elements through MK and shape of Kc.An energy function associated with these conditions is given by

Ih(Th,Tc)=1

3 ∑K∈Th

|K|det(MK)12(

tr((F′K)

−1M−1K (F′

K)−T))2

+4

3 ∑K∈Th

|K|det(MK)12

(

det(F′K)det(MK)

12

)−2, (4.4)

which is actually a Riemann sum of a continuous functional developed in [14] based onmesh equidistribution and alignment.

Note that Ih(Th,Tc) is a function of the vertices ξ j =(ξ1j ,ξ2

j ), j=1,··· ,Nv, of the com-

putational mesh Tc and the vertices xj =(xj,yj), j=1,··· ,Nv, of the physical mesh Th. A

straight way of solving the minimization problem is to take Tc as Tc and then solvethe minimization problem of Ih(Th,Tc) for the new physical mesh T

n+1h . However,

Ih(Th,Tc) is highly nonlinear in ξ j =(ξ1j ,ξ2

j ), j=1,··· ,Nv. The fact that M is a function of

x and thus MK is a function of the coordinates of the physical vertices makes the situa-tion more difficult. Here, we adopt an indirect approach, i.e., to take Th as T n

h , minimizeIh(T

nh ,Tc) with respect to Tc, and then obtain the new physical mesh using the rela-

tion between T nh and newly obtained Tc. The minimization is carried out by integrating

the mesh equation which is defined as the gradient system of the energy function (theMMPDE approach), viz.,

dξ j

dt=−det(M(xj))

12

τ

(∂Ih(Tn

h ,Tc)

∂ξξξ j

)T, j=1,··· ,Nv, (4.5)

where ∂Ih/∂ξ j is considered as a row vector, τ > 0 is a parameter used to adjust theresponse time of mesh movement to the changes in M. Using the notion of scalar-by-matrix differentiation [16], we can rewrite (4.5) as

dξ j

dt=

det(M(xj))12

τ ∑K∈ωj

|K|vKjK

, j=1,··· ,Nv, (4.6)

where ωj is the element patch associated with the vertex xj, jK is the local index of xj in

K, and vKjK

is the local velocity contributed by K. Denote the edge matrices of K and Kc by


EK = [xK1 −xK

0 , xK2 −xK

0 ] and EKc = [ξK1 −ξK

0 , ξK2 −ξK

0 ], respectively. Let J =(F′K)

−1 = EKc E−1K

and define

G(J,det(J))=1

3det(MK)

12 (tr(JM−1

K JT))2+4

3det(MK)

12

(

det(J)

det(MK)12

)2

.

It is not difficult [16] to find the derivatives of G with respect to J and det(J) as

∂G

∂J=

4

3det(MK)

12 (tr(JM−1

K JT))M−1K JT,

∂G

∂det(J)=

8

3det(MK)

− 12 det(J).

(4.7)

Then, the local velocities are expressed as

[

(vK1 )

T

(vK2 )

T

]

=−E−1K

∂G

∂J− ∂G

∂det(J)

det(EKc)

det(EK)E−1

Kc, vK

0 =−vK1 −vK

2 . (4.8)

Note that the velocities for the boundary nodes need to be modified properly. For ex-ample, the velocities for the corner vertices should be set to be zero. For other boundaryvertices, the velocities should be modified such that they only slide along the boundaryand do not move out of the domain. Specifically, we assume that the domain boundarycan be expressed as ψ(ξ)= 0. Consider a boundary vertex ξ j on the boundary which is

not a corner and denote the mesh equation (4.5) bydξ j

dt =uj. This equation can be modifiedinto

dξ j

dt=uj−

(∇ψ·uj)∇ψ

|∇ψ|2 ,

which satisfies ∇ψ· dξ j

dt =0 and makes the node slide on the boundary.The mesh equation (4.6) can be integrated from tn to tn+1, starting with the reference

computational mesh Tc as the initial mesh. The obtained new mesh is denoted by T n+1c .

Note that T nh is kept fixed during the integration and it and T n+1

c form a correspondence,

i.e., T nh =Φh(T

n+1c ). Then the new physical mesh T

n+1h is defined as T

n+1h = Φh(Tc)

which can be readily computed using linear interpolation.To conclude this section we would like to point out that a number of other moving

mesh methods have been developed in the past and there is a vast literature in the area.The interested reader is referred to the books/review articles [1–3,18,37] and some recentarticles [4, 30, 38, 43].

5 Numerical examples

In this section we present numerical results obtained with the moving mesh DG methoddescribed in the previous sections for a number of one- and two-dimensional exam-


ples for the RTE. Unless otherwise stated, we use in the discrete-ordinate approxima-tion the Gauss-Legendre P8 and the Legendre-Chebyshev P8-T8 rules for one- and two-dimensional problems, respectively, and take the final time as T=0.1. By trial and error,we have found that the time step size ∆t = 10−3 is sufficiently small so that the spatialerror dominates the total error in the computation. For mesh movement, we take τ=0.1for Example 5.3 and τ = 0.01 for the others. For the cases having an exact solution, theerror in the computed solution is measured in the (global) L1, L2, and L∞ norm, i.e.,

∫ T

0‖eh(·,t)‖L1 dt,

∫ T

0‖eh(·,t)‖L2 dt,

∫ T

0‖eh(·,t)‖L∞ dt.

Example 5.1 (A discontinuous test for the one-dimensional RTE simulating the absorbing-s-cattering model). We take extinction coefficient σt=1000, σs =1, c=3.0×108, and

q(x,µ,t)=σt sin(2π(tanh(Rx)+5µt))+10πµ

ccos(2π(tanh(Rx)+5µt))

+2πµRcos(2π(tanh(Rx)+5µt))(1−tanh2(Rx))

+σs

20πt(cos(2π(tanh(Rx)+5t))−cos(2π(tanh(Rx)−5t)))+a(σt−σs),

where a=2 and R=200. The initial condition is I(x,µ,0)= sin(2πtanh(Rx))+a and theboundary conditions are

I(−1,µ,t)=sin(2π(−tanh(R)+5µt))+a, 0<µ≤1, 0< t≤T,

I(1,µ,t)=sin(2π(tanh(R)+5µt))+a, −1≤µ<0, 0< t≤T.

The exact solution of this example is I(x,µ,t)=sin(2π(tanh(Rx)+5µt))+a. The meshtrajectories for the P2-DG method with a moving mesh of N=80 are shown in Fig. 1. Themoving mesh solution (N = 80) in the direction µ=−0.5255 is compared with the fixed

-1 -0.5 0 0.5 1

x

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t

Figure 1: Example 5.1. The mesh trajectories are obtained with the P2-DG method with a moving mesh ofN=80.


-1 -0.5 0 0.5 1

x

1

1.5

2

2.5

3

I(x,

3)

exactfixed N=80moving N=80

(a) MM N=80, FM N=80

-0.05 0 0.05

x

1

1.5

2

2.5

3

I(x,

3)


(b) Close view of (a) near x=0

-1 -0.5 0 0.5 1

x

1

1.5

2

2.5

3

I(x,

3)


(c) MM N=80, FM N=1280

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

x

1

1.5

2

2.5

3

I(x,

3)


(d) Close view of (c) near x=0

Figure 2: Example 5.1. The solution in the direction µ =−0.5255 obtained with the P2-DG method with amoving mesh of N=80 is compared with those obtained with fixed meshes of N=80 and N=1280. MM andFM stand for moving mesh and fixed mesh, respectively.

mesh solutions obtained with N = 80 and N = 1280 in Fig. 2. Similar results are shownin Fig. 3 for the angular direction µ= 0.9603. These results show that the moving meshsolution (N=80) is more accurate than those with fixed meshes of N=80 and N=1280.

The error in the L1 and L2 norm is shown in Fig. 4 for both P1-DG and P2-DG methodswith fixed and moving meshes. It can be seen that both fixed and moving meshes lead toalmost the same order of convergence for relatively large N, i.e., 2nd order for P1-DG and3rd-order for P2-DG. However, a moving mesh always produces more accurate solutionsthan a fixed mesh of the same size for this example.

To show the efficiency of the methods, we plot in Fig. 5 the L1 norm of the error againstthe CPU time measured in seconds on a Thinkpad T440 with Matlab 2017a. One can seethat moving mesh (MM) P1-DG (resp. P2-DG) is more efficient than fixed mesh (FM) P1-DG (resp. P2-DG) in the sense that the former leads to a smaller error than the latter fora fixed amount of the CPU time. Moreover, when N>50, MM (resp. FM) P2-DG is more


-1 -0.5 0 0.5 1

x

1

1.5

2

2.5

3

I(x,

8)


(a) MM N=80, FM N=80

-0.05 0 0.05

x

1

1.5

2

2.5

3

I(x,

8)


(b) Close view of (a) near x=0

-1 -0.5 0 0.5 1

x

1

1.5

2

2.5

3

I(x,

8)


(c) MM N=80, FM N=1280

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

x

1

1.5

2

2.5

3

I(x,

8)


(d) Close view of (c) near x=0

Figure 3: Example 5.1. The solution in the direction µ=0.9603 obtained with the P2-DG method with a movingmesh of N=80 is compared with those obtained with fixed meshes of N=80 and N=1280.

1.5 2 2.5 3 3.5

log10

(N)

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

log

10(e

rror

)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 2

(a) P1-DG

1.5 2 2.5 3 3.5

log10

(N)

-7

-6

-5

-4

-3

-2

log

10(e

rror

)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 3

(b) P2-DG

Figure 4: Example 5.1. The L1 and L2 norm of the error with moving and fixed meshes.


1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

log10

(CPU time)

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

log

10(L

1-e

rror

)

FM-P1FM-P2MM-P1MM-P2

Figure 5: Example 5.1. The L1 norm of the error is plotted against the CPU time.

efficient than MM (resp. FM) P1-DG. Thus, a moving mesh improves the computationalefficiency and the quadratic DG method has better efficiency than the linear one on bothfixed and moving meshes when N is sufficiently large.

Example 5.2 (A discontinuous example of the one-dimensional RTE for the absorbing-s-cattering model). In this example, we take σs =1, c=3.0×108, and

σt =

1, for 0≤ x<0.2,

900, for 0.2≤ x<0.6,

90, for 0.6≤ x≤1,

q(x,µ,t)=

100e−t, for 0≤ x<0.2,

1, for 0.2≤ x<0.6,

1000e3t , for 0.6≤ x≤1.

The initial condition is I(x,µ,0)=15x and the boundary conditions are given by

I(0,µ,t)=0, for 0<µ≤1, 0< t≤T,

I(1,µ,t)=15+2t, for −1≤µ<0, 0< t≤T.

The solution of this problem has two sharp layers. Since its analytical form is unavailablefor comparison purpose we take the numerical solution obtained with the P2-DG methodwith a fixed mesh of N = 20000 as the reference solution. The mesh trajectories for theP2-DG method with a moving mesh of N = 80 are shown in Fig. 6. The moving meshsolution (N=80) in the direction µ=−0.1834 is compared with the fixed mesh solutionsobtained with N = 80 and N = 1280 in Fig. 7. Similar results are shown in Fig. 8 for theangular direction µ=0.1834. The results show that the moving mesh solution (N=80) ismore accurate than those with fixed meshes of N=80 and N=1280.


0 0.2 0.4 0.6 0.8 1

x

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t

Figure 6: Example 5.2. The mesh trajectories are obtained with the P2-DG method with a moving mesh ofN=80.

0 0.2 0.4 0.6 0.8 1

x

0

10

20

30

40

50

60

70

80

I(x,

4)

referencefixed N=80moving N=80

(a) MM N=80, FM N=80

0.18 0.19 0.2 0.21

x

0

2

4

6

8

10

12I(

x,4)

0.58 0.6 0.62

x

0

2

4

6

8

10

12

14

16

I(x,

4)

(b) Close view of (a) near x=0.2 and x=0.6

0 0.2 0.4 0.6 0.8 1

x

0

10

20

30

40

50

60

70

80

I(x,

4)


(c) MM N=80, FM N=1280

0.199 0.2 0.201

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

I(x,

4)

0.597 0.598 0.599 0.6 0.601

x

0

2

4

6

8

10

12

14

16

I(x,

4)

(d) Close view of (c) near x=0.2 and x=0.6

Figure 7: Example 5.2. The solution in the direction µ =−0.1834 obtained with the P2-DG method with amoving mesh of N=80 is compared with those obtained with fixed meshes of N=80 and N=1280.


0 0.2 0.4 0.6 0.8 1

x

0

10

20

30

40

50

60

70

80I(

x,5)


(a) MM N=80, FM N=80

0.18 0.2 0.22 0.24

x

0

10

20

30

40

50

60

70

80

I(x,

5)

0.58 0.6 0.62 0.64

x

0

2

4

6

8

10

12

14

16

I(x,

5)


0 0.2 0.4 0.6 0.8 1

x

0

10

20

30

40

50

60

70

80

I(x,

5)

referencefixed N

x=1280

moving Nx=80

(c) MM N=80, FM N=1280

0.199 0.2 0.201 0.202 0.203

x

0

10

20

30

40

50

60

70

80

I(x,

5)

0.599 0.6 0.601

x

0

1

2

3

4

5

6

7

8

I(x,

5)


Figure 8: Example 5.2. The solution in the direction µ=0.1834 obtained with the P2-DG method with a movingmesh of N=80 is compared with those obtained with fixed meshes of N=80 and N=1280.

Example 5.3 (An accuracy test for the two-dimensional RTE for the absorbing-scatteringmodel). In this example, we take σt=22000, σs =1, c=3.0×108, and

q(x,y,ζ,η,t)=et

(

−2π(ζ+η)(ζ2+η2)cos3(π

2(x+y)

)

sin(π

2(x+y)

)

+((1

c+σt

)

(ζ2+η2)− 2

3σs

)

cos4(π

2(x+y)

)

+(1

c+σt−σs

)

)

.

The initial condition is I(x,y,ζ,η,0)=(ζ2+η2)cos4(π2 (x+y))+1 and the boundary condi-

tions are given by

I(x,0,ζ,η,t)= et

(

(ζ2+η2)cos4(π

2x)

+1

)

, η>0,

I(x,1,ζ,η,t)= et

(

(ζ2+η2)cos4(π

2(x+1)

)

+1

)

, η<0,


2 2.5 3 3.5 4 4.5

log10

(N)

-12

-11

-10

-9

-8

-7

-6

-5

-42l

og10

(err

or)

fixed: L1

moving: L1

fixed: Lmoving: L

slope = - 2

(a) P1-DG

2 2.5 3 3.5 4 4.5

log10

(N)

-16

-14

-12

-10

-8

-6

2log

10(e

rror

)

fixed: L1

moving: L1

fixed: Lmoving: L

slope = - 3

(b) P2-DG

Figure 9: Example 5.3. The L1 and L∞ norm of the error with moving and fixed meshes.

I(0,y,ζ,η,t)= et

(

(ζ2+η2)cos4(π

2y)

+1

)

, ζ>0,

I(1,y,ζ,η,t)= et

(

(ζ2+η2)cos4(π

2(1+y)

)

+1

)

, ζ<0.

The exact solution of this problem is I(x,y,ζ,η,t) = et(

(ζ2+η2)cos4(π2 (x+y))+1

)

. The

error in the L1 and L∞ norm is plotted in Fig. 9 for the P1-DG and P2-DG methods withfixed and moving meshes. It can be seen that both fixed and moving meshes lead tocomparable results and the same convergence order (2nd for P1-DG and 3rd for P2-DG),which is consistent with the theoretical prediction.

Example 5.4 (A discontinuous example of the two-dimensional RTE for the transparentmodel). In this test, we solve the two-dimensional RTE (2.2) with σt=0, σs=0, c=3.0×108 ,q=0, ζ=0.3, and η=0.5. The computational domain is (0,1)×(0,1). The initial conditionis

I(x,y,ζ,η,0)=

0, for y<ηζ x,

cos6(

π2 y)

, otherwise.

The boundary conditions are

I(0,y,ζ,η,t)=cos6(π

2y)

cos10(t), I(x,0,ζ,η,t)=0.

The exact solution of this example is

I(x,y,ζ,η,t)=

0, for y<ηζ x,

cos6(

π2 (y−

ηζ x))

cos10(t− xcζ ), otherwise.


(a) Radiative intensity on MM N=1600

0 0.2 0.4 0.6 0.8 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

(b) MM N=1600 at t=0.1

(c) Radiative intensity on FM N=1600 (d) Radiative intensity on FM N=57600

Figure 10: Example 5.4. The radiative intensity contours (and mesh) at t=0.1 obtained by the P2-DG methodwith a moving mesh of N=1600 and fixed meshes of N=1600 and N=57600.

Notice that only a single angular direction is chosen in this example and the integral termin (2.2) is not involved.

The radiative intensity contours obtained with a moving mesh of N=1600 and fixedmeshes of N = 1600 and N = 57600 are shown in Fig. 10. In Fig. 11, we compare theradiative intensity cut along the line y= 0.495 for the moving mesh of N = 1600 and thefixed meshes of N=1600 and N=57600. The results show that the moving mesh solution(N=1600) is better than that with the fixed mesh of N=1600 and is comparable with thatwith the fixed mesh of N=57600.

The error in the L1 and L2 norm is shown in Fig. 12 for the P1-DG and P2-DG meth-ods with fixed and moving meshes. It is worth pointing out that we cannot expect P1-DGand P2-DG can achieve their optimal order for this problem since the solution is discon-tinuous. One can see from the figure that both fixed and moving meshes lead to almostthe same convergence order. The order of P1-DG is about 0.72 in L1 norm and 0.36 in L2

norm while that of P2-DG is about 0.81 in L1 norm and 0.41 in L2 norm. Moreover, the


0 0.2 0.4 0.6 0.8 1

x

0

0.2

0.4

0.6

0.8

1

I


(a) MM N=1600, FM N=1600

0 0.2 0.4 0.6 0.8 1

x

0

0.2

0.4

0.6

0.8

1

I


0.26 0.3 0.34

x

0

0.2

0.4

0.6

0.8

1

I

(b) MM N=1600, FM N=57600

Figure 11: Example 5.4. The comparison of the radiative intensity cut along the line y=0.495 obtained by the

P2-DG method with a moving mesh of N=1600 and fixed meshes of N=1600 and N=57600.

2.5 3 3.5 4 4.5 5 5.5

log10

(N)

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

2log

10(e

rror

)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 1

(a) P1-DG

2.5 3 3.5 4 4.5 5 5.5

log10

(N)

-8.5

-8

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.52l

og10

(err

or)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 1

(b) P2-DG


figure shows that a moving mesh produces more accurate solutions than a fixed mesh ofthe same number of elements for this example.

Example 5.5 (A discontinuous example of the two-dimensional RTE for the purely ab-sorbing model). In this example, we choose σt =1, σs =0, c=3.0×108, q=0, ζ =0.4, andη=0.9. The computational domain is (0,1)×(0,1). The initial condition is

I(x,y,ζ,η,0)=

tanh(500(x−0.5))+1, for y<ηζ x,

1, otherwise.

The boundary conditions are

I(0,y,ζ,η,t)= e(y2 t), I(x,0,ζ,η,t)= tanh(500(x−0.5))+1.



0 0.2 0.4 0.6 0.8 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

(b) MM N=6400 at t=0.1


Figure 13: Example 5.5. The radiative intensity contours (and mesh) at t=0.1 obtained by the P2-DG methodwith a moving mesh of N=6400 and fixed meshes of N=6400 and N=102400.

The exact solution of this example is

I(x,y,ζ,η,t)=

(tanh(500(x− ζη y−0.5))+1)e−

σtη y

, for y<ηζ x,

e

(

(y− ηζ x)2(t− x

cζ )−σtζ x)

, otherwise,

which exhibits a discontinuity along with y= ηζ x and a sharp layer along with x= ζ

η y+0.5.

Like the previous example, only a single angular direction is chosen in this example andthe integral term in (2.2) is not involved.

The radiative intensity contours obtained with the P2-DG method with a movingmesh of N = 6400 and fixed meshes of N = 6400 and N = 102400 are shown in Fig. 13.In Fig. 14, the radiative intensity cut along the line y=0.495 is compared for moving andfixed meshes. The advantage of using a moving mesh is clear. The error in the L1 and L2

norm is plotted as a function of N in Fig. 15 for fixed and moving meshes. The P1-DGmethod shows an order of about 0.87 in L1 norm and 0.57 in L2 norm while P2-DG has


0 0.2 0.4 0.6 0.8 1

x

0

0.2

0.4

0.6

0.8

1

1.2

I


(a) MM N=6400, FM N=6400

0.2 0.25 0.3

x

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

I

0.7 0.75 0.8 0.85

x

0

0.2

0.4

0.6

0.8

1

1.2

I


0 0.2 0.4 0.6 0.8 1

x

0

0.2

0.4

0.6

0.8

1

1.2

I


(c) MM N=6400, FM N=102400

0.18 0.2 0.22 0.24 0.26

x

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

I

0.68 0.7 0.72 0.74 0.76 0.78

x

0

0.2

0.4

0.6

0.8

1

1.2

I


Figure 14: Example 5.5. The radiative intensity cut along the line y=0.495 obtained a moving mesh of N=6400is compared with those obtained with fixed meshes of N=6400 and N=102400.

2.5 3 3.5 4 4.5 5 5.5

log10

(N)

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

2log

10(e

rror

)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 1

(a) P1-DG

2.5 3 3.5 4 4.5 5 5.5

log10

(N)

-9

-8

-7

-6

-5

-4

-3

2log

10(e

rror

)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 1

(b) P2-DG



1 1.5 2 2.5 3 3.5 4 4.5

log10

(CPU time)

-4.2

-4

-3.8

-3.6

-3.4

-3.2

-3

-2.8

-2.6

-2.4

-2.2

log

10(L

1-e

rror

)

FM-P1FM-P2MM-P1MM-P2

Figure 16: Example 5.5. The L1 norm of the error is plotted against the CPU time.

an order of 1.2 in L1 norm and 0.73 in L2 norm for both fixed and moving meshes. Amoving mesh produces more accurate solutions than a fixed mesh of the same numberof elements for this example.

To show the efficiency of the methods for this two-dimensional example, we plot theL1 norm of the error against the CPU time in Fig. 16. We can see that the error is smallerfor moving mesh P1-DG (resp. P2-DG) than fixed mesh P1-DG (resp. P2-DG) for a fixedamount of the CPU time. Moreover, the better efficiency of a higher-order method ismore obvious in this example than the one-dimensional Example 5.1: Fixed mesh P2-DGis nearly equally or more efficient than moving mesh P1-DG.

Example 5.6 (A steep transition layer example of the two-dimensional RTE for the ab-sorbing-scattering model). In this example, the parameters are taken as σt=10000, σs=1,c=3.0×108, and

q(x,y,ζ,η,t)=et

(

(1

c+σt

)(

a−tanh(

R(x2+y2−√

2(1−ζ2−η2))

)

−2R(ζx+ηy)(

1−tanh2(R(x2+y2−√

2(1−ζ2−η2))

)

+

√2σs

2R

(

ln(

cosh(R(√

2−x2−y2)))

−ln(

cosh(R(x2+y2)))

)

−σsa

)

,

where R=200 and a=10. The initial condition is

I(x,y,ζ,η,0)=(

a−tanh(

R(x2+y2−√

2(1−ζ2−η2))

)

and the boundary conditions are

I(x,0,ζ,η,t)= et(

a−tanh(

R(x2−√

2(1−ζ2−η2))

)

, η>0,

I(x,1,ζ,η,t)= et(

a−tanh(

R(x2+1−√

2(1−ζ2−η2))

)

, η<0,



0 0.2 0.4 0.6 0.8 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

(b) MM N = 6400 at t=0.1


Figure 17: Example 5.6. The radiative intensity contours in the direction Ω=(−0.2578,−0.1068) (and mesh)

at t = 0.1 obtained by P2-DG method with a moving mesh of N = 6400 and fixed meshes of N = 6400 andN=57600.

I(0,y,ζ,η,t)= et(

a−tanh(

R(y2−√

2(1−ζ2−η2))

)

, ζ>0,

I(1,y,ζ,η,t)= et(

a−tanh(

R(1+y2−√

2(1−ζ2−η2))

)

, ζ<0.

This problem has the exact solution

I(x,y,ζ,η,t)= et(

a−tanh(

R(x2+y2−√

2(1−ζ2−η2))

)

,

for which the location of the steep transition layers changes with the angular variableΩ=(ζ,η). The radiative intensity contours for the directions Ω=(−0.2578,−0.1068) andΩ = (0.7860,0.3256) are shown in Figs. 17 and 18, respectively, for the P2-DG methodwith moving and fixed meshes. It can be seen that the elements of the moving mesh areconcentrated in the regions of the sharp transition layers in the radiative intensities for all



0 0.2 0.4 0.6 0.8 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

(b) MM N = 6400 at t=0.1


Figure 18: Example 5.6. The radiative intensity contours in the direction Ω=(0.7860,0.3256) (and mesh) at

t=0.1 obtained by P2-DG method with a moving mesh of N=6400 and fixed meshes of N=6400 and N=57600.

angular directions. (The mesh shows four layers while only two layers are shown in theintensity contours in Figs. 17 and 18. The figures for the intensities for other directionsare omitted to save space.) It can be seen that the moving mesh (N = 6400) provides abetter resolution of the layers than the fixed mesh of N=6400 and is comparable with thefixed mesh of N=57600.

The error in the L1 and L2 norm is plotted in Fig. 19 as a function of N. The conver-gence order is similar for both fixed and moving meshes, i.e., the order of P1-DG is about1.7 in L1 norm and 1.6 in L2 norm and that of P2-DG is 2.2 in both L1 and L2 norm.

Example 5.7 (A steep transition layer example of the two-dimensional RTE for the ab-sorbing-scattering model). In this final example, we take σt =33, σs =3, and c=3.0×108.The computational domain is (−1,1)×(−1,1). Define

C0(x,y)= tanh

(

R(

x2+y2− 1

8

)

)

,


2.5 3 3.5 4 4.5 5

log10

(N)

-9

-8

-7

-6

-5

-4

-32l

og10

(err

or)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 1

(a) P1-DG

2.5 3 3.5 4 4.5 5

log10

(N)

-10

-9

-8

-7

-6

-5

-4

-3

2log

10(e

rror

)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 1

(b) P2-DG


C1(x,y)= tanh

(

R(

(x−0.5)2+(y−0.5)2− 1

8

)

)

,

C2(x,y)= tanh

(

R(

(x−0.5)2+(y+0.5)2− 1

8

)

)

,

C3(x,y)= tanh

(

R(

(x+0.5)2+(y−0.5)2− 1

8

)

)

,

C4(x,y)= tanh

(

R(

(x+0.5)2+(y+0.5)2− 1

8

)

)

,

where R=200. The source term is taken as

q(x,y,ζ,η,t)

=et(ζ2+η2)

(

(1

c+σt

)(

5a−4

∑i=0

Ci(x,y))

−R(

2(ζx+ηy)(1−C0(x,y)2)

+(ζ(2x−1)+η(2y−1))(1−C1(x,y)2)+(ζ(2x−1)+η(2y+1))(1−C2(x,y)2)

+(ζ(2x+1)+η(2y−1))(1−C3(x,y)2)+(ζ(2x+1)+η(2y+1))(1−C4(x,y)2))

)

− 2

3etσs

(

5a−4

∑i=0

Ci(x,y))

,

where a = 2. The initial condition is I(x,y,ζ,η,0) = (ζ2+η2)(

5a−∑4i=0Ci(x,y)

)

and theboundary conditions are

I(x,0,ζ,η,t)= et(ζ2+η2)(

5a−4

∑i=0

Ci(x,0))

, η>0,



-1 -0.5 0 0.5 1

x

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y

(b) MM N = 6400 at t=0.1


Figure 20: Example 5.7. The radiative intensity contours in the direction Ω=(0.2578,0.1068) (and mesh) at

t=0.1 obtained by P2-DG method with a moving mesh of N=6400 and fixed meshes of N=6400 and N=57600.

I(x,1,ζ,η,t)= et(ζ2+η2)(

5a−4

∑i=0

Ci(x,1))

, η<0,

I(0,y,ζ,η,t)= et(ζ2+η2)(

5a−4

∑i=0

Ci(0,y))

, ζ>0,

I(1,y,ζ,η,t)= et(ζ2+η2)(

5a−4

∑i=0

Ci(1,y))

, ζ<0.

The exact solution of the problem is

I(x,y,ζ,η,t)= et(ζ2+η2)(

5a−4

∑i=0

Ci(x,y))

,

which exhibits a sharp layer of five-ring shape, independent of ζ and η. The radiativeintensity contours in the direction Ω = (0.2578,0.1068) are shown in Fig. 20 for the P2-


-1 -0.5 0 0.5 1

x

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6I


(a) MM N=6400, FM N=6400

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x

0.55

0.56

0.57

0.58

0.59

0.6

I


(b) Close view of (a) near x∈ (−0.35,0.35)

-1 -0.5 0 0.5 1

x

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

I


(c) MM N=6400, FM N=57600

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

x

0.55

0.56

0.57

0.58

0.59

0.6

I


(d) Close view of (c) near x∈ (−0.35,0.35)

Figure 21: Example 5.7. The radiative intensity in the direction Ω=(0.2578,0.1068) cut along the line y=0.8xobtained a moving mesh of N = 6400 is compared with those obtained with fixed meshes of N = 6400 andN=57600.

DG method with moving and fixed meshes. In Fig. 21, the radiative intensity cut alongthe line y= 0.8x for the direction Ω=(0.2578,0.1068) is compared for moving and fixedmeshes. The error in L1 and L2 norm is plotted as a function of N in Fig. 22.

6 Conclusions

In the previous sections an adaptive moving mesh DG method has been presented for thenumerical solution of the radiative transfer equation. The RTE is an integro-differentialequation modeling the conservation of photons. It involves an integral term in the angu-lar directions while being hyperbolic in space. The challenges for its numerical solutioninclude the needs to handle with its high dimensionality, the presence of the integral


2.5 3 3.5 4 4.5 5

log10

(N)

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

2log

10(e

rror

)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 1

(a) P1-DG

2.5 3 3.5 4 4.5 5

log10

(N)

-9

-8

-7

-6

-5

-4

-3

-2

2log

10(e

rror

)

fixed: L1

moving: L1

fixed: L2

moving: L2

slope = - 1

(b) P2-DG


term, and the development of discontinuities and sharp layers in its solution along spa-tial directions. In the current work, the RTE is discretized first in the angular directionswith the discrete ordinate method and then with a DG method in space on a movingmesh. The mesh is moved adaptively using the MMPDE strategy to provide better res-olution of sharp layers or discontinuities and thus better efficiency. The source iterationis used to avoid the coupling of the radiative intensities for all angular directions in theintegral term.

A selection of one- and two-dimensional examples have been presented to demon-strate the accuracy and efficiency of the method. It has been shown that the method isable to automatically concentrate the mesh points in regions of discontinuities or steeptransition layers and is more efficient than its fixed mesh counterpart. Particularly, thecombination of the metric tensors for radiative intensities for different angular directionsinto a single metric tensor using the matrix intersection seems to work well for the testedproblems with single or multiple sharp layers and discontinuities. Interestingly, the re-sults also show that the quadratic DG method has better efficiency than the linear DG forboth fixed and moving meshes.

It should be pointed out that we have not considered positivity-preserving limiters[39, 42] nor nonoscillatory limiters such as TVB limiter [7, 9], the WENO limiter [33, 40],or HWENO limiter [34,41] in the current work. One may observe that localized spuriousoscillations occur in numerical solutions containing discontinuities. How to combine lim-iters with our moving mesh DG method for the RTE or more general integro-differentialequations will be an interesting research topic for the near future. Other future workwill include extending the method to the numerical solution of the RTE coupled with theEuler equations, the material equation or the energy equation for real situations.


Acknowledgments

This research was partly supported by China NSAF grant U1630247, Science ChallengeProject, No. TZ2016002, and NSFC grant 11871111.

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