# A moving mesh finite difference method for equilibrium radiation diffusion equations

## Abstract

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 multi-material, 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.

- Authors:

- Department of Mathematics, College of Science, China University of Mining and Technology, Xuzhou, Jiangsu 221116 (China)
- Department of Mathematics, University of Kansas, Lawrence, KS 66045 (United States)
- School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005 (China)

- Publication Date:

- OSTI Identifier:
- 22465664

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 298; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCURACY; CONCENTRATION RATIO; CONVERGENCE; DIFFUSION EQUATIONS; ENERGY DENSITY; EQUILIBRIUM; FINITE DIFFERENCE METHOD; MESH GENERATION; NONLINEAR PROBLEMS; RADIATION TRANSPORT

### Citation Formats

```
Yang, Xiaobo, E-mail: xwindyb@126.com, Huang, Weizhang, E-mail: whuang@ku.edu, and Qiu, Jianxian, E-mail: jxqiu@xmu.edu.cn.
```*A moving mesh finite difference method for equilibrium radiation diffusion equations*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2015.06.014.

```
Yang, Xiaobo, E-mail: xwindyb@126.com, Huang, Weizhang, E-mail: whuang@ku.edu, & Qiu, Jianxian, E-mail: jxqiu@xmu.edu.cn.
```*A moving mesh finite difference method for equilibrium radiation diffusion equations*. United States. doi:10.1016/J.JCP.2015.06.014.

```
Yang, Xiaobo, E-mail: xwindyb@126.com, Huang, Weizhang, E-mail: whuang@ku.edu, and Qiu, Jianxian, E-mail: jxqiu@xmu.edu.cn. Thu .
"A moving mesh finite difference method for equilibrium radiation diffusion equations". United States. doi:10.1016/J.JCP.2015.06.014.
```

```
@article{osti_22465664,
```

title = {A moving mesh finite difference method for equilibrium radiation diffusion equations},

author = {Yang, Xiaobo, E-mail: xwindyb@126.com and Huang, Weizhang, E-mail: whuang@ku.edu and Qiu, Jianxian, E-mail: jxqiu@xmu.edu.cn},

abstractNote = {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 multi-material, 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.},

doi = {10.1016/J.JCP.2015.06.014},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 298,

place = {United States},

year = {2015},

month = {10}

}