A semi-adaptive compact splitting method for the numerical solution of 2-dimensional quenching problems

Abstract This article studies a semi-adaptive compact Peaceman–Rachford splitting method for solving two-dimensional nonlinear reaction–diffusion equations with singular source terms. While an adaption is utilized in the temporal direction, uniform grids are considered in the space. It is shown that the compact scheme is stable and convergent when its dimensional Courant numbers are within the frame of a window determined by the given spatial domain. Though such a window implies a considerable restriction on the decomposed compact scheme, the new computational strategy acquired is highly efficient and reliable for a variety of applications. Numerical examples are given to illustrate our conclusions and indicate that the method constructed is effective in determining key singularity characteristics such as the quenching time and critical domain.

[1]  Compact schemes in application to singular reaction-diffusion equations , 2012, Proceedings of the 2012 44th Southeastern Symposium on System Theory (SSST).

[2]  Jens Lang,et al.  An adaptive Rothe method for nonlinear reaction-diffusion systems , 1993 .

[3]  Sining Zheng,et al.  Asymptotic estimates to quenching solutions of heat equations with weighted absorptions , 2010, Asymptot. Anal..

[4]  Hideo Kawarada,et al.  On Solutions of Initial-Boundary Problem for ut=uxx+\frac{1}{1−u} , 1973 .

[5]  W. Olmstead,et al.  The influence of two moving heat sources on blow-up in a reactive-diffusive medium , 2000 .

[6]  A. Wouwer,et al.  Adaptive Method of Lines , 2001 .

[7]  J. Verwer,et al.  A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines , 1990 .

[8]  C. Y. Chan,et al.  Parabolic Quenching for Nonsmooth Convex Domains , 1994 .

[9]  Qin Sheng,et al.  Adaptive decomposition finite difference methods for solving singular problems—A review , 2009 .

[10]  P. Henrici.  Discrete Variable Methods in Ordinary Differential Equations , 1962 .

[11]  Qin Sheng,et al.  A compound adaptive approach to degenerate nonlinear quenching problems , 1999 .

[12]  Matthew A. Beauregard,et al.  An adaptive splitting approach for the quenching solution of reaction-diffusion equations over nonuniform grids , 2013, J. Comput. Appl. Math..

[13]  Hong Cheng,et al.  Solving Degenerate Reaction-Diffusion Equations via Variable Step Peaceman-Rachford Splitting , 2004, SIAM J. Sci. Comput..

[14]  Jianping Zhu,et al.  An efficient high‐order algorithm for solving systems of reaction‐diffusion equations , 2002 .

[15]  Andrew Acker,et al.  Remarks on quenching , 1988, Documenta Mathematica.

[16]  Hideo Kawarada,et al.  On Solutions of Initial-Bound ary Problem 1 for ut=uxx+- l-u , 1975 .

[17]  Robert D. Russell,et al.  Adaptivity with moving grids , 2009, Acta Numerica.

[18]  Ping Lin,et al.  A splitting moving mesh method for reaction-diffusion equations of quenching type , 2006, J. Comput. Phys..

[19]  A moving mesh method in multiblock domains with application to a combustion problem , 1999 .

[20]  Edward H. Twizell,et al.  Chaos-free numerical solutions of reaction-diffusion equations , 1990, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[21]  P. C. Kong,et al.  Channel flow of a viscous fluid in the boundary layer , 1997 .

[22]  W. Olmstead,et al.  Blow-up in a reactive-diffusive medium with a moving heat source , 2002 .

[23]  T. K. Boni.  Extinction for discretizations of some semilinear parabolic equations , 2001 .

[24]  E. M. Elsayed,et al.  DIFFERENCE EQUATIONS , 2001 .

[25]  T. K. Boni.  On quenching of solutions for some semilinear parabolic equations of second order , 2000 .

[26]  A. Peterson,et al.  Difference Equations: An Introduction with Applications , 2000 .

[27]  Qin Sheng,et al.  An adaptive grid method for degenerate semilinear quenching problems , 2000 .

[28]  Nejla Nouaili.  A Liouville theorem for a heat equation and applications for quenching , 2011 .

[29]  Howard A. Levine,et al.  Quenching, nonquenching, and beyond quenching for solution of some parabolic equations , 1989 .

[30]  H. Ockendon.  Channel flow with temperature-dependent viscosity and internal viscous dissipation , 1979 .