High-order Fully Implicit Scheme and Multigrid Method for Two-dimensional Semilinear Diffusion Reaction Equations

doi:10.19596/j.cnki.1001-246x.8042

Abstract

Abstract: A finite difference method is used for high-order numerical solution of two-dimensional unsteady semilinear diffusion reaction equation. The spatial derivative term is discretized by a fourth-order compact difference formula, and the time derivative term is discretized by a fourth-order backward Euler formula. An unconditionally stable high-order five-level fully implicit scheme is proposed. Truncation error of the scheme is O(τ⁴+τ²h²+h⁴), that is, the time and space have fourth-order accuracy. In calculation of start-up steps, the first, second and third time levels are discretized by Crank-Nicolson method. Richardson extrapolation formula was used to extrapolate startup time accuracy to the fourth-order. A multigrid method based on the scheme is established, which accelerates convergence speed of the algebraic equations on each time level and improves computational efficiency. Finally, accuracy, stability and efficiency of the scheme and multigrid approach are verified with numerical experiments.

Key words: semilinear diffusion reaction equation, high-oder fully implicit scheme, compact scheme, multigrid method, Richardson extrapolation

CLC Number:

O241.82

ZHANG Lin, GE Yongbin. High-order Fully Implicit Scheme and Multigrid Method for Two-dimensional Semilinear Diffusion Reaction Equations[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(3): 307-319.

References

[1] FRANK-KAMENETSKII D A. Diffusion and heat transfer in chemical kinetics[M]. New York:Plenum Press, 1969.
[2] AMES W F. Nonlinear partial differential equations in engineering[M]. New York:Academic Press, 1965.
[3] WANG Y M. Error and extrapolation of a compact LOD method for parabolic differential equations[J]. Journal of Computational and Applied Mathematics, 2011, 235(2):1336-1382.
[4] QUARTERONI A, VALLI A. Numerical approximation of partial differential equations[M]. New York:Springer-Verlag, 1997.
[5] ARAÚJO A, BARBEIRO S, SERRANHO P. Convergence of finite difference schemes for nonlinear complex reaction-diffusion processes[J]. SIAM Journal on Numerical Analysis, 2015, 53:228-250.
[6] WANG Y M, WANG J. A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equaionts[J]. Computers & Mathematics with Applications, 2011, 62:2434-2451.
[7] SUN Z Z. Numerical methods of partial differential equations[M]. Beijing:Science Press, 2005.
[8] LV G X, SUN S K. A finite directional difference meshless method for diffusion equations[J]. Chinese Journal of Computational Physics, 2015, 32:649-661.
[9] ZHANG B L, SU X M. An alernating segment Crank-Nicholson scheme for parabolic equation[J]. Chinese Journal of Computational Physics, 1995, 12:115-120.
[10] FERNANDES R I, BIALECKI B, FAIRWEATHER G. An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction-diffusion systems on evolving domains[J]. Journal of Computational Physics, 2015, 299:561-580.
[11] LI D, WANG J. Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system[J]. Journal of Scientific Computing, 2017, 72:892-915.
[12] ZHANG S H, LIANG D. A Strang-type alternating segment domain decomposition method for two-dimensional parabolic equations[J]. Chinese Journal of Computational Physics, 2018, 35(4):413-428.
[13] WANG Y M, GUO B Y. A monotone compact implicit scheme for nonlinear reaction-diffusion equations[J]. Journal of Computational Mathematics, 2008, 26:123-148.
[14] 吴宏伟. 二维半线性反应扩散方程的交替方向隐格式[J]. 计算数学, 2008, 30(4):349-360.
[15] WU F Y, CHENG X J, LI D F, DUAN J Q. A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction-diffusion equations[J]. Computers & Mathematics with Applications, 2018, 75:2835-2850.
[16] ZHOU H, WU Y J, TIAN W Y. Extrapolation algorithm of compact ADI approximation for two-dimensional parabolic equation[J]. Applied Mathematics and Computation, 2012, 219(6):2875-2884.
[17] 马廷福, 金涛, 葛永斌. 二维扩散反应方程的三次样条高精度加权隐格式差分格式及其多重网格方法[J]. 兰州理工大学学报, 2013, 36(3):141-146.
[18] 田振夫. 扩散反应方程的三次样条高阶差分格式[J]. 湖北大学学报(自然科学版), 1997, 2:111-115.
[19] 葛永斌, 田振夫, 吴文权. 二维抛物型方程的高精度多重网格解法[J]. 应用数学, 2003, 16(2):13-18.
[20] GE Y B, CAI Z Q. High-order compact difference schemes and adaptive method for singular degenerate diffusion-reaction equations[J]. Chinese Journal of Computational Physics, 2017, 34:309-319.
[21] STRIKWERDA J C. Finite difference schemes and partial differential equations[M]. New York:Chyampan & Hall, 1989.
[22] STRIKWERDA J C. High order accurate schemes for incompressible viscous flow[J]. International Journal for Numerical Methods in Fluids, 1997, 24(7):715-734.
[23] WANG T, WANG Y M. A higher-order compact LOD method and its extrapolations for nonhomogeneous parabolic differential equations[J]. Applied Mathematics and Computation, 2014, 237:512-530.
[24] BRANDT A. Multi-level adaptive solution to boundary-value problems[J]. Mathematics of Computation, 1977, 31:330-390.
[25] QIU X C, XUE Y Y. Application analysis of multi-grid method for solving the nuclear reactor diffusion equation[J]. Chinese Journal of Computational Physics, 1991, 8(4):387-394.
[26] WESSELING P W. An introduction to multigrid methods[M]. Chichester:Wiley, 1992.
[27] 葛永斌, 吴文权, 田振夫. 二维波动方程的高精度隐格式及其多重网格算法[J]. 厦门大学学报(自然科学版), 2003, 42(6):691-696.
[28] FISHER R A. The wave of advance of advantageous genes[J]. Ann Eugenics, 1937, 355:7.
[29] CHAFEE N, INFANTE E F. A bifurcation problem for a nonlinear partial differential equation of parabolic type[J]. Applicable Analysis, 1974, 4(1):21.