Finite volume element methods for nonequilibrium radiation diffusion equations

SUMMARY Nonequilibrium radiation diffusion problems are described by the coupled radiation diffusion and material conduction equations. Because of the highly nonlinear, strong discontinuous, and tightly coupled phenomena, solving this kind of problems is a challenge. We construct two finite volume element schemes for the equations. One of them is monotone on many kinds of meshes, which is proved theoretically and verified by numerical tests. The other one is hard to satisfy the monotonicity, but this defect can be corrected by different repair techniques. Numerical results show that these new methods are practical and efficient on distorted meshes.Copyright © 2013 John Wiley & Sons, Ltd.

[1]  Gordon L. Olson,et al.  Efficient solution of multi-dimensional flux-limited nonequilibrium radiation diffusion coupled to material conduction with second-order time discretization , 2007, J. Comput. Phys..

[2]  Dana A. Knoll,et al.  Temporal Accuracy of the Nonequilibrium Radiation Diffusion Equations Applied to Two-Dimensional Multimaterial Simulations , 2006 .

[3]  Daniil Svyatskiy,et al.  Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes , 2007, J. Comput. Phys..

[4]  Prateek Sharma,et al.  Preserving monotonicity in anisotropic diffusion , 2007, J. Comput. Phys..

[5]  Junliang Lv,et al.  L2 error estimate of the finite volume element methods on quadrilateral meshes , 2010, Adv. Comput. Math..

[6]  Ivar Aavatsmark,et al.  Monotonicity of control volume methods , 2007, Numerische Mathematik.

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

[8]  D. A. Knoll,et al.  New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion , 2003 .

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

[10]  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..

[11]  Raphaël Loubère,et al.  The repair paradigm: New algorithms and applications to compressible flow , 2006 .

[12]  Zhiqiang Sheng,et al.  Monotone Finite Volume Schemes of Nonequilibrium Radiation Diffusion Equations on Distorted Meshes , 2009, SIAM J. Sci. Comput..

[13]  Jiwoong Choi,et al.  A multiscale MDCT image-based breathing lung model with time-varying regional ventilation , 2013, J. Comput. Phys..

[14]  Yunqing Huang,et al.  Some new discretization and adaptation and multigrid methods for 2-D 3-T diffusion equations , 2007, J. Comput. Phys..

[15]  R. Glowinski,et al.  A multigrid preconditioner and automatic differentiation for non-equilibrium radiation diffusion problems , 2005 .

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

[17]  Vidar Thomée,et al.  On the existence of maximum principles in parabolic finite element equations , 2008, Math. Comput..

[18]  Zhiqiang Sheng,et al.  Discrete maximum principle based on repair technique for diamond type scheme of diffusion problems , 2012 .

[19]  Mikhail Shashkov,et al.  The repair paradigm and application to conservation laws , 2004 .

[20]  Ludmil T. Zikatanov,et al.  A monotone finite element scheme for convection-diffusion equations , 1999, Math. Comput..

[21]  Guangwei Yuan,et al.  Analysis and construction of cell-centered finite volume scheme for diffusion equations on distorted meshes , 2009 .

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

[23]  Zhiqiang Sheng,et al.  A Nine Point Scheme for the Approximation of Diffusion Operators on Distorted Quadrilateral Meshes , 2008, SIAM J. Sci. Comput..