A moving mesh finite difference method for equilibrium radiation diffusion equations

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.

[1]  Michael L. Norman,et al.  Adaptive-mesh radiation hydrodynamics—II. The radiation and fluid equations in relativistic flows , 1984 .

[2]  Robert D. Russell,et al.  Adaptive Moving Mesh Methods , 2010 .

[3]  William J. Rider,et al.  An efficient nonlinear solution method for non-equilibrium radiation diffusion , 1999 .

[4]  S. McCormick,et al.  The fast adaptive composite grid (FAC) method for elliptic equation , 1986 .

[5]  Weizhang Huang,et al.  Variational mesh adaptation II: error estimates and monitor functions , 2003 .

[6]  Jim E. Morel,et al.  Numerical analysis of time integration errors for nonequilibrium radiation diffusion , 2007, J. Comput. Phys..

[7]  Stephen F. McCormick,et al.  Multilevel adaptive methods for partial differential equations , 1989, Frontiers in applied mathematics.

[8]  L. Margolin,et al.  On balanced approximations for time integration of multiple time scale systems , 2003 .

[9]  Weizhang Huang,et al.  A high dimensional moving mesh strategy , 1998 .

[10]  Michael L. Norman,et al.  Adaptive-mesh radiation hydrodynamics. I - The radiation transport equation in a completely adaptive coordinate system. II - The radiation and fluid equations in relativistic flows , 1984 .

[11]  Yue Jingyan,et al.  Progress in Numerical Methods for Radiation Diffusion Equations , 2009 .

[12]  William J. Rider,et al.  Nonlinear convergence, accuracy, and time step control in nonequilibrium radiation diffusion , 2001 .

[13]  R. Bowers,et al.  Numerical Modeling in Applied Physics and Astrophysics , 1991 .

[14]  Weizhang Huang.  Practical aspects of formulation and solution of moving mesh partial differential equations , 2001 .

[15]  Changhua Yu,et al.  Finite volume element methods for nonequilibrium radiation diffusion equations , 2013 .

[16]  William J. Rider,et al.  A Multigrid Newton-Krylov Method for Multimaterial Equilibrium Radiation Diffusion , 1999 .

[17]  J. R. Cash,et al.  Diagonally Implicit Runge-Kutta Formulae with Error Estimates , 1979 .

[18]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[19]  Guangwei Yuan,et al.  Picard-Newton Iterative Method with Time Step Control for Multimaterial Non-Equilibrium Radiation Diffusion Problem , 2011 .

[20]  William J. Rider,et al.  Physics-Based Preconditioning and the Newton-Krylov Method for Non-equilibrium Radiation Diffusion , 2000 .

[21]  Michael Pernice,et al.  Solution of Equilibrium Radiation Diffusion Problems Using Implicit Adaptive Mesh Refinement , 2005, SIAM J. Sci. Comput..

[22]  D. Mihalas,et al.  Foundations of Radiation Hydrodynamics , 1985 .

[23]  Weizhang Huang,et al.  Moving Mesh Methods Based on Moving Mesh Partial Differential Equations , 1994 .

[24]  Weizhang Huang,et al.  Variational mesh adaptation: isotropy and equidistribution , 2001 .

[25]  Robert B. Lowrie,et al.  A comparison of implicit time integration methods for nonlinear relaxation and diffusion , 2004 .

[26]  Xiao-Chuan Cai,et al.  One‐level Newton–Krylov–Schwarz algorithm for unsteady non‐linear radiation diffusion problem , 2004, Numer. Linear Algebra Appl..

[27]  Luis Chacón,et al.  Cost-effectiveness of fully implicit moving mesh adaptation: A practical investigation in 1D , 2006, J. Comput. Phys..

[28]  Weizhang Huang,et al.  The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and Lubrication-type equations , 2012, J. Comput. Phys..

[29]  A. Dvinsky.  Adaptive grid generation from harmonic maps on Reimannian manifolds , 1991 .