High-order Compact Difference Schemes and Adaptive Method for Singular Degenerate Diffusion-Reaction Equations

Abstract

Abstract: We established a high-order compact scheme on nonuniform grids for singular degenerate diffusion reaction equation. The scheme is second order accuracy in time and third to fourth order accuracy in space. A grid adaptive method is established by using equidistribution principle. Finally,a numerical example with exact solution validates reliability and accuracy of the method. One-dimentional blow up problem was solved with the method.

Key words: singular and degenerate diffusion-reaction equation, adaptive method, nonuniform grid, high order compact scheme, blow up

CLC Number:

O241.82

GE Yongbin, CAI Zhiquan. High-order Compact Difference Schemes and Adaptive Method for Singular Degenerate Diffusion-Reaction Equations[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2017, 34(3): 309-319.

References

[1] BRANCH M C, BECKSTEAD M W, LITZINGER T A, et al. Nonsteady combustion mechanisms of advanced solid propellants[R]. Annual Technical Report, 94-05, Center for Combustion and Environmental Research, University of Colorado, Boulder, CO (1994).
[2] GHAFOURIAN A, HUYN C, JOHNSON P, et al. Liquid rocket combustion instability[R]. Research Report, 90-02, Center for Combustion and Environmental Research, University of Colorado, Boulder, CO (1990).
[3] GOVAERTS W. Seeking for reference on combustion problems[J]. Numer Anal,1997, 17(5).
[4] FUJITA H. On the blowing up of solutions of the Cauchy problem for u_t=Δu+u^1+α[J]. J Fac Sci Univ Tokyo Sec A, 1996, 16:105-113.
[5] KAWARADA H. On solutions of initial-boundary value problem for u_t=u_xx+1/(1-u)[J]. Publ Res Inst Math Sci, 1975, 10:729-736.
[6] KHALIQ A Q M, TWIZELL E H, VOSS D A. On parallel algorithms for semidiscretized parabolic partial differential equations based on subdiagonal pade approximations[J]. Numer Methods Partial Differential Eq,1993, 9:107-116.
[7] MOONEY J W. A numerical method for accurate critical length estimation in singular quenching problems[J]. WSSIA, 1995, 4:505-516.
[8] MOONEY J W. An implicit algorithm for iterating to quenching times in degenerate nonlinear parabolic problems[J]. Dynamic Sys Appl, 1996, 5:539-551.
[9] LANG J. Two-dimensional fully adaptive solutions of reaction-diffusion equations[J]. Appl Numer Math,1995, 18:223-240.
[10] LANG J,WALTER A. An adaptive method for nonlinear reaction-diffusion systems[J]. Appl Numer Math, 1993, 13:135-146.
[11] SHENG Q. Adaptive decomposition finite difference methods for solving singular problems[J]. Frontiers Math China, 2009, 4:599-626.
[12] SHENG Q, AGARWAL R. Nonlinear variation of parameter methods for summary difference equations in several independent variables[J]. J Appl Math Comput,1994, 61:39-60.
[13] SHENG Q, CHENG H. An adaptive grid method for degenerate semilinear quenching problems[J]. Computers Math Appl, 2000,39:57-71.
[14] SHENG Q, KHALIQ A Q M. A compound adaptive approach to degenerate nonlinear quenching problems[J]. Numer Methods Partial Differential Eq, 1999, 15:29-47.
[15] SHENG Q, KHALIQ A Q M. Linearly implicit adaptive schemes for singular reaction-diffusion equations[M].Chapter 9, Adaptive Method of Lines, CRC Press, London and New York, 2001.
[16] SHENG Q, KHALIQ A Q M. Modified arc-length adaptive algorithms for degenerate reaction-diffusion equations[J]. Appl Math Comput, 2002, 126:279-297.
[17] SHENG Q, KHALIQ A Q M. A Revisit of the semi-adaptive method for singular degenerate reaction-diffusion equations[J]. East Asian Journal on Applied Mathematics, 2012, 2:185-203.
[18] MATTHEW A BEAUREGARD, QIN Sheng. A fully adaptive approximation for quenching-type reaction-diffusion equations over circular domains[J]. Numer Methods Partial Differential Eq, 201430:472-489.