Positivity-preserving moving mesh scheme for two-step reaction model in two dimensions ☆

Abstract In this paper, we have proposed a moving mesh scheme for the two-step chemical reaction model in two dimensions. This moving mesh scheme consists of two independent parts: governing equations evolution and mesh-redistribution. The second-order finite volume scheme is used in the governing equations evolution part. In the mesh-redistribution part, the mesh points are first redistributed, and then the cell averages of the conservative variables are remapped onto the new mesh in a conservative way. In addition, the positivity-preserving improvements are shown for the two parts. The positivity-preserving thresholds guarantee the average value positive, and then an improved implementation is given to make each cell value positive. Several numerical experiments are carried out to demonstrate the accuracy and effectiveness of the proposed method.

[1]  Ke Chen.  Error Equidistribution and Mesh Adaptation , 1994, SIAM J. Sci. Comput..

[2]  Jianqiang Han,et al.  An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics , 2007, J. Comput. Phys..

[3]  Jiang Zonglin,et al.  Half-Cell Law of Regular Cellular Detonations , 2008 .

[4]  Xiangxiong Zhang,et al.  Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes , 2011, Journal of Scientific Computing.

[5]  Tao Tang,et al.  Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws , 2003, SIAM J. Numer. Anal..

[6]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[7]  Xiangxiong Zhang,et al.  A minimum entropy principle of high order schemes for gas dynamics equations , 2011, Numerische Mathematik.

[8]  Bernard Parent,et al.  Positivity-preserving high-resolution schemes for systems of conservation laws , 2012, J. Comput. Phys..

[9]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[10]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes , 2010, J. Comput. Phys..

[11]  Chi-Wang Shu,et al.  On positivity preserving finite volume schemes for Euler equations , 1996 .

[12]  Xiaobo Yang,et al.  A Moving Mesh WENO Method for One-Dimensional Conservation Laws , 2012, SIAM J. Sci. Comput..

[13]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[14]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[15]  Xiangxiong Zhang,et al.  On maximum-principle-satisfying high order schemes for scalar conservation laws , 2010, J. Comput. Phys..

[16]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[17]  J. Ning,et al.  A Pseudo Arc-Length Method for Numerical Simulation of Shock Waves , 2014 .

[18]  Ruo Li,et al.  Moving Mesh Discontinuous Galerkin Method for Hyperbolic Conservation Laws , 2006, J. Sci. Comput..

[19]  W. Fickett,et al.  Flow Calculations for Pulsating One‐Dimensional Detonations , 1966 .

[20]  Cheng Wang,et al.  Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations , 2012, J. Comput. Phys..

[21]  Gang Wang,et al.  An improved CE/SE scheme for numerical simulation of gaseous and two-phase detonations , 2010 .

[22]  John M. Stockie,et al.  A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws , 2000, SIAM J. Sci. Comput..

[23]  Xiangxiong Zhang,et al.  Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms , 2011, J. Comput. Phys..

[24]  S. Taki,et al.  NUMERICAL SIMULATION ON THE ESTABLISHMENT OF GASEOUS DETONATION , 1984 .

[25]  Huazhong Tang,et al.  An adaptive moving mesh method for two-dimensional relativistic magnetohydrodynamics , 2012 .