1. Introduction
The subject capacity of entropy is huge. In thermodynamics, entropy can refer to a physical quantity that can be measured by the change of heat. In many computational problems, the nonequilibrium radiation diffusion equations play a very important role. These real physical applications include the inertial confinement fusion, Z-pinch experiments, and astrophysical problems. When the thermodynamic equilibrium between the radiation field and the material is not reached, a set of coupled radiation diffusion and material conduction equations are needed to simulate the transfer and exchange of energy. With the multiple materials, strongly nonlinear and tightly coupled with the problems, the accurate numerical method is essential to simulate the diffusion processes of these applications.
In recent years, various numerical algorithms [
1,
2,
3] were proposed to obtain reliable numerical solutions of nonequilibrium radiation diffusion equations. The Jacobian-free Newton–Krylov method is given in [
4,
5] to solve the equations. In [
6], the preconditioned Jacobian-free Newton–Krylov methods are considered, and it investigates in [
7] a minor improvement to the operator-splitting preconditioner. The time integration methods are presented in [
8,
9,
10], and their efficiency and accuracy are further considered in [
11]. The second-order time discretization method for the coupled multidimensional flux-limited nonequilibrium radiation diffusion and material conduction equations is studied in [
12]. Moreover, two different time-step control methods were given in [
13]. Nevertheless, these methods cannot be used to solve the equations on distorted meshes. The radiation diffusion problems are often combined with other physical processes. For example, in Lagrangian radiation hydrodynamics, the solution is based on hydrodynamic mesh, which is often distorted by fluid motion. Therefore, the constructed numerical method should be able to simulate the radiation diffusion problem on severely distorted meshes.
Positivity-preserving is one of the key requirements for constructing the discrete scheme of nonequilibrium radiation diffusion equations. Against the background of heat transfer, if the discrete scheme does not satisfy this property, it may violate the entropy constraint of the second law of thermodynamics, and it will affect the numerical accuracy of the scheme and bring spurious oscillations. In [
14,
15], the positivity-preserving finite volume schemes for nonequilibrium radiation diffusion problems are constructed, and the positivity-preserving property of the schemes is also derived. With the applied interpolation method, these methods are only applicable to the problems with geometric restrictions. Moreover, the interpolation algorithms are not positivity-preserving. In [
16], two finite volume element methods are presented. It was proved that one of them is monotonic, and another is positivity-preserving under some postprocessing techniques. However, these methods only can be used for the meshes with geometric restrictions. In [
17], the authors propose a positivity-preserving finite point method for the nonequilibrium radiation diffusion equations. However, this method is not conservative, and it can only be used for the uniform mesh. In [
18], a unified gas-kinetic scheme (UGKS) for a coupled system of radiative transport and material heat conduction with different diffusive limits was constructed, which also only considers uniform mesh.
In this paper, we propose a new positivity-preserving finite volume scheme for the nonequilibrium radiation diffusion equations based on [
19,
20] This numerical scheme is fixed stencil, local conservative, and positivity-preserving. The numerical scheme has the characteristics of fixed stencil, local conservative, positivity-preserving and so on. The cell-vertexes are used to define auxiliary unknowns. Therefore, vertex interpolation algorithm will be used to obtain the value of the auxiliary unknown. With the distorted meshes, the existing vertex positivity-preserving interpolation algorithms usually have significant accuracy loss. However, this scheme dose not need the interpolation method to be a positivity-preserving one. Besides, the decomposition of normal vectors on unstructured meshes is highly efficient. In practical applications, when the classical Picard iteration method solves the final nonlinear algebraic system, its convergence rate may be very slow. To speed up the convergence of nonlinear iterations, the Anderson acceleration method is used here.
The outline of this paper is as follows. In
Section 2, some notations of mesh and the nonequilibrium radiation diffusion equations are presented. In
Section 3, we present the construction of the new positivity-preserving finite volume scheme. Some theoretical analysis of discrete schemes is in
Section 4. Then, in
Section 5 we present some numerical experiments to illustrate the features of the scheme. We end our presentation in
Section 6 with some conclusions.
2. Model and Notations
We consider a system of multimaterial nonequilibrium radiation diffusion coupled with material energy balance equation, which is defined in the domain
with the reflection boundary conditions. The equations are written as:
where
E is defined as the radiation energy density;
D is the radiation diffusion coefficient;
T is defined as the material temperature;
is the corresponding material conduction coefficient; and t is represented as time.
The energy exchange is controlled by the photon absorption cross-section:
where
Z represents the atomic mass number, and its value is related to the material.
We first define the radiation diffusion coefficient without flux limiter as:
However, in regions with strong radiation energy gradients, the model may be unphysical, with the propagation velocity of a radiation wave front in a vacuum faster than the speed of light. To avoid this unphysical phenomenon, we add a limiting term to the diffusion coefficient and adopt such a flux-limited diffusion coefficient:
The material conduction coefficient
is taken as the following form:
where the constant
is chosen to be
.
Throughout this article, we employ the following notations and define the discretization of a finite volume scheme on as , where
denotes a family of partitions of the domain into nonoverlapping mesh cells, and . For , , and denote the cell boundary, diameter (the maximum distance between any two points in K) and measure, respectively. Besides, is the mesh sizes;
is a finite family of disjoint edges in such that for , is a line segment whose measure is defined as . Let and . For , there exists a subset of such that . denotes the unit vector normal to outward to K;
is a set of points defined as cell centers, where ;
is also a set of point, where represents the set of vertices of cell K, where are oriented in a counter clockwise direction. is the number of vertex for cell K.
Let this problem time discrete with the uniform time step . and are defined as the approximate solutions of E and T at the cell center as the primary variables, respectively. The approximation of solutions E and T at the vertex point are defined as the auxiliary variables and denoted as and .
According to the idea of finite volume framework, the above Equations (
1) and (2) are integrated into each control volume
K, and we can then obtain:
where
and
are the fluxes
and
, respectively.
(resp.
) is the one-sided flux that approximates
(resp.
) using only the information of cell
K.
5. Numerical Results
In this section, we present some numerical examples to investigate the accuracy and efficiency of the new positive-preserving finite volume scheme on distorted meshes.
The GMRES linear solver is used to solve the linear systems with stopping tolerance
. The stopping tolerance of nonlinear iteration is taken to be
. The discrete solution errors and flux errors are investigated in the discrete
norms, which are defined by the following expressions:
where
is the measure associated with
,
is the numerical flux, and the analytical flux
is evaluated by the midpoint rule. Besides, we define the
-norm of solution to compare the numerical results on different meshes. Let:
The energy conservation error is a criterion to judge the precision of the scheme. Define the total energy:
The energy conservation error is
, where
is the total energy at initial time, and
is the total energy at final time. Besides, the following notations are used for the numerical tests
- -
itn: average number of linear iterations;
- -
nitn: average number of nonlinear Picard iterations;
- -
: minimal value of the numerical solution E;
- -
: minimal value of the numerical solution T.
5.1. Accuracy Test
In this test, we will consider the parabolic equation on the unit square
as:
with the full Dirichlet boundary condition. The diffusion tensor
D is taken as:
The exact solution is chosen to be
:
This problem is run up to time
with the time step
. This problem is solved on three type meshes as in
Figure 3, such that the nodes of unstructured mesh located on line
are distorted only in the
y-direction. In
Figure 4, the solution and flux errors are graphically depicted as log–log plots of the discrete
norm errors versus the mesh size
h. The results demonstrate that the scheme has second (respectively first)-order convergence rate with the solution (resp. flux).
5.2. Results without Flux Limiter
Mousseau and Knoll present an interesting two-dimensional multimaterial test problem that has a central blast wave moving out around two square obstacles. The Equations (
1) and (2) are solved on the
mesh grid as
Figure 3. The background material uses
, while the obstacles use
. These obstacles are located at:
All four walls are insulated with respect to radiation diffusion and material conduction:
and
The initial energy distribution has a Gaussian peak near the origin:
The initial material temperature is taken to be
. The initial radiation spreads out and flows around the obstacles. This problem is simulated with the final time
and time step
. The contour of radiation energy density and material temperature at time
are presented in
Figure 5 and
Figure 6, respectively. We can see that the contours on random mesh and triangular mesh accord with that on uniform mesh. In
Table 1, the minimal value and
-norm of numerical solution on different mesh are similar. Moreover, the minimal values of the numerical solution
E and
T are positive. The energy conservation error on different mesh is almost machine precision. The average number of nonlinear iterations in one time step and the average number of linear iterations per nonlinear iteration are also shown in
Table 1. These numerical results indicate that our scheme is positivity-preserving, conservative, and robust in solving this problem.
The results of Anderson acceleration for the Picard iteration method are also presented in
Table 2. The total number of nonlinear iterations is reduced significantly by the Anderson acceleration. The reasonable choice for this problem is
, since the total number of nonlinear iterative times is not observably decreasing for
and
. From these numerical results, we know that the positivity-preserving scheme is robust and accurate in simulating this problem on distorted meshes.
5.3. Results with Flux Limiter
In this numerical example, we consider simulating the Equations (
1) and (2) with flux limiter on the
mesh as
Figure 3. The initial and boundary conditions are given as the above numerical example. This problem is run up to
, and the time step is taken as
.
The contours of radiation energy density and material temperature are shown in
Figure 7 and
Figure 8, respectively. The contours on unstructured meshes also accord with that on uniform mesh. Moreover, no spurious oscillation can be seen on the distorted meshes. In
Table 3, some numerical results of the radiation energy density and material temperature are presented. The energy conservation errors of this problem are almost machine-precise. The
-norm and minimal value of the numerical solution on distorted meshes closely approximate that on uniform mesh. The average number of nonlinear iterations and linear iterations illustrate the positivity-preserving scheme is efficient. These numerical results of the scheme are very close to the numerical solution presented in [
14,
15,
16].
The performance of the Anderson acceleration for the Picard iteration method is also considered. In
Table 4, the total number of nonlinear iterations with the different values of
m are presented, which shows the effectiveness of Anderson acceleration method. These results illustrate that
is a better choice for this test.