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 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.

A finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes

“…We provide a brief description of this equation. See [27,36] for a more detailed discussion. Consider the following nonlinear parabolic partial differential equation:…”

Moving mesh finite difference solution of non-equilibrium radiation diffusion equations

“…The shape, size, and orientation of mesh elements are controlled through a monitor function [8] defined through the Hessian of the energy density. A similar moving mesh FD method has been developed in [37] for equilibrium radiation diffusion equations, and the current work can be considered as a generalization of [37]. However, this generalization is non-trival.…”

“…However, this generalization is non-trival. Unlike [37], we now need to deal with a system of two coupled equations for the energy density and material temperature. The diffusion coefficients depend on both the energy density and material temperature and it is more sensitive to treat diffusion numerically.…”

“…Furthermore, it is more delicate to preserve the solution positivity. Like [37], we use here the cutoff strategy to maintain the positivity in the computed solutions. It has been shown in [21] that the strategy retains the accuracy and convergence order of FD approximation for parabolic PDEs.…”

“…Such complex local structures make mesh adaptation an indispensable tool for use to improve the efficiency in the numerical solution of radiation diffusion equations because the number of mesh points can be prohibitively large when a uniform mesh is used. Research of radiation diffusion has attracted considerable attentions from engineers and scientists [4,17,18,23,24,25,26,27,29,30,31,33,35,36,37,38].…”