Journal of Computational Physics 298 (2015) 661–677

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Na * ht 00Journal of Computational Physics www.elsevier.com/locate/jcp moving mesh finite difference method for equilibrium adiation diffusion equations✩ iaobo Yang a, Weizhang Huang b, Jianxian Qiu c,∗ epartment of Mathematics, College of Science, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China epartment of Mathematics, University of Kansas, Lawrence, KS 66045, USA chool of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific mputing, Xiamen University, Xiamen, Fujian 361005, China r t i c l e i n f o a b s t r a c t ticle history: ceived 9 February 2015 cepted 20 June 2015 ailable online 29 June 2015 ywords: oving mesh method uilibrium radiation diffusion equations ediction–correction eezing coefficient nnegativity toff o-level mesh movement

An efficient moving mesh finite difference method is developed for the numerical solution of equilibrium radiation diffusion equations in two dimensions. The method is based on the moving mesh partial differential equation approach and moves the mesh continuously in time using a system of meshing partial differential equations. The mesh adaptation is controlled through a Hessian-based monitor function and the so-called equidistribution and alignment principles. Several challenging issues in the numerical solution are addressed.

Particularly, the radiation diffusion coefficient depends on the energy density highly nonlinearly. This nonlinearity is treated using a predictor–corrector and lagged diffusion strategy. Moreover, the nonnegativity of the energy density is maintained using a cutoff method which has been known in literature to retain the accuracy and convergence order of finite difference approximation for parabolic equations. Numerical examples with multimaterial, multiple spot concentration situations are presented. Numerical results show that the method works well for radiation diffusion equations and can produce numerical solutions of good accuracy. It is also shown that a two-level mesh movement strategy can significantly improve the efficiency of the computation. © 2015 Elsevier Inc. All rights reserved.

Introduction

Radiation diffusion plays an important role in a variety of physical applications such as inertially confined fusion, comstion simulation, and atmospheric dynamics. When photon mean free paths are much shorter than characteristic length ales, a diffusion approximation can be used to describe the radiation penetrating from a hot source to a cold medium. This ffusion approximation forms a highly nonlinear diffusion coefficient and gives a sharp hot wave steep front (often referred as a Marshak wave). Solutions near this steep front can vary dramatically in a very short distance. Such complex local lution structures make radiation diffusion an excellent example for using mesh adaptation methods because the number mesh points can be prohibitively large when a uniform mesh is used.

Research was supported in part by National Natural Science Foundation of China through grants 91230110, 11328104, 11426214, and 2014QNB35, and tional Science Foundation (USA) through grant DMS-1115118.

Corresponding author.

E-mail addresses: xwindyb@126.com (X. Yang), whuang@ku.edu (W. Huang), jxqiu@xmu.edu.cn (J. Qiu). tp://dx.doi.org/10.1016/j.jcp.2015.06.014 21-9991/© 2015 Elsevier Inc. All rights reserved. 662 X. Yang et al. / Journal of Computational Physics 298 (2015) 661–677 th ot e. nu st (G on of flu in an m te no co ac si ap en co no co di st du is ea ea no in ed st fo no sp in us th co

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Fi 2. fo ra w ciGenerally speaking, there are two types of radiation diffusion equations, equilibrium and non-equilibrium systems. When e energy density E satisfies the relation E = T 4, where T is temperature, the system is called in an equilibrium state and herwise in a non-equilibrium state. Radiation diffusion has attracted considerable attention from researchers in the past; g., see [3,16,21,23–25,28–30]. For example, foundations of radiation hydrodynamics can be found in the book [21] while merical techniques for radiation diffusion and transport are addressed systematically in the book [3]. Rider et al. [25] udy multi-material equilibrium radiation diffusion equations and propose a class of algorithms with the Newton–Krylov

MERS) method preconditioned by a multi-grid method as a special example. Ovtchinnikov and Cai [23] study a parallel e-level Newton–Krylov–Schwarz algorithm for an unsteady nonlinear radiation diffusion problem.

On the other hand, there are only a few published works that have employed mesh adaptation for the numerical solution radiation diffusion equations. Winkler et al. [26,27] use an adaptive moving mesh method to solve radiation diffusion and id equations in one dimension. Unfortunately, their method has difficulty with mesh crossing and cannot be extended multi-dimensions. Lapenta and Chacón [16] study an equilibrium radiation diffusion equation in one dimension using adaptive moving mesh method. Pernice and Philip [24] use the AMR method (a structured adaptive mesh refinement ethod) to solve two-dimensional equilibrium radiation diffusion equations. They employ a fully implicit scheme to ingrate the partial differential equations (PDEs), JFNK (Jacobian-free Newton–Krylov) [14,15,22,25] to solve the resulting nlinear algebraic equations, and the FAC (Fast Adaptive Composite) preconditioner [19,20] to precondition the implicit efficient matrix. Their numerical results show that the method can capture the fronts of Marshak waves and have good curacy for problems with smooth initial solutions.

The objective of this paper is to study the finite difference (FD) solution of two-dimensional equilibrium radiation diffuon equations using an adaptive moving mesh method. The method is based on the so-called moving mesh PDE (MMPDE) proach [10] with which the mesh is moved continuously in time using an MMPDE. The latter is defined as the gradit flow equation of a meshing functional formulated based on mesh equidistribution and alignment and taking into full nsideration of the shape, size and orientation of mesh elements [7]. The method is combined with treatments of high nlinearity and preservation of solution nonnegativity of the equations. The high nonlinearity comes from the diffusion efficient. We use a coefficient-freezing predictor–corrector procedure to linearize the PDEs. More specifically, at the prection stage the diffusion coefficient is calculated using the energy density at the previous time step while at the correction age the coefficient is calculated using the energy density obtained at the prediction stage. This predictor–corrector procere is known to be comparable to the Beam and Warming linearization method in terms of accuracy and stability [17]. It also easy and efficient to implement since it contains only two steps of the lagged diffusion computation. Note that for ch stage we only need to solve linear PDEs so there is no need for nonlinear iteration. Moreover, the procedure allows an sy and effective dealing of negative values occurring in the computed energy density. Radiation diffusion equations admit nnegative energy densities. It is crucial for numerical approximation to preserve this property. Excessive negative values the computed solution not only introduce unphysical oscillations but also can cause the computation to exit unexpectly. We use a cutoff strategy with which negative values in the computed energy density are replaced with zero after each age. It has been shown in [18] that the cutoff strategy retains the accuracy and convergence order of FD approximation r parabolic PDEs.