High Accurate Non-Equidlstant Method for Singular Perturbation Reaction-Diffusion Problem

CAI Xin; CAI Dan-lin; WU Rui-qian; XIE Kang-he

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Article Navigation > Applied Mathematics and Mechanics > 2009 > 30(2): 171-178

CAI Xin, CAI Dan-lin, WU Rui-qian, XIE Kang-he. High Accurate Non-Equidlstant Method for Singular Perturbation Reaction-Diffusion Problem[J]. Applied Mathematics and Mechanics, 2009, 30(2): 171-178.

Citation:

CAI Xin, CAI Dan-lin, WU Rui-qian, XIE Kang-he. High Accurate Non-Equidlstant Method for Singular Perturbation Reaction-Diffusion Problem[J]. Applied Mathematics and Mechanics, 2009, 30(2): 171-178.

Citation:

CAI Xin, CAI Dan-lin, WU Rui-qian, XIE Kang-he. High Accurate Non-Equidlstant Method for Singular Perturbation Reaction-Diffusion Problem[J]. Applied Mathematics and Mechanics, 2009, 30(2): 171-178.

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High Accurate Non-Equidlstant Method for Singular Perturbation Reaction-Diffusion Problem

1.
School of Science, Zhejiang University of Science and Technology, Hangzhou 310027, P. R. China;

Received Date: 2008-06-25
Rev Recd Date: 2008-12-04
Publish Date: 2009-02-15

Abstract

Abstract

Singular pertubation reaction-diffusion problem with Dirichlet boundary condition is considered. this is a multi-scale problem. The presence of small parameter leads to bomdary Dyer phenomena on both sides of region. Non-equidistant finite difference method wag presented according to the property of boundary layer. The region was divided into the inner botmdary layer region and the outside botutdaty layer regiart according to transition point of Shishkin. The step length is equidistant on the outside bowdaty layer region. The step length is gradually increased on the inner boiutdacy layer region such that half of the step length is digerent firm each other. Tnutcation error was estimated. The new method is stable and uniform convergence with order higher than 2. Finally, numerical results were given, which are in agreement with the theoretical result.
- singular perturbation,
- reaction-diffusion,
- uniform convergence,
- highly accurate,
- non-equidistant

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References(14)

References

[1]	Farrell P, Hegarty A F, Miller J J H,et al. Robust Computational Techniques for Boundary Layers[M]. Boca Raton: Chapman and Hall/CRC, 2000.
[2]	Miller J J H, O'Riordan E, Shishkin G I.Fitted Numerical Methods for Singular Perturbation Problems[M].Singapore: World Scientific, 1996.
[3]	蔡新. 具有周期边界的守恒型方程的守恒型差分格式[J]. 应用数学和力学, 2001, 22(10):1092-1096.
[4]	CAI Xin, LIU Fa-wang. Uniform convergence difference schemes for singularly perturbed mixed boundary problems[J].Journal of Computational and Applied Mathematics,2004,166(1):31-54. doi: 10.1016/j.cam.2003.09.038
[5]	CAI Xin, LIU Fa-wang. A Reynolds uniform scheme for singularly perturbed parabolic differential equation[J].ANZIAM J,2007,47(5):633-648.
[6]	Kellogg R B, Tsan A. Analysis of some difference approximations for a singular perturbation problem without turning points[J].Math Comp,1978,26(12):1025-1039.
[7]	Bakhvalov N S. On the optimization of methods for boundary-value problems with boundary layers[J].USSR Computational Mathematics and Mathematical Physics,1969,9(4):139-166.
[8]	Jayakumar J. Improvement of numerical solution by boundary value technique for singularly perturbed one dimensional reaction diffusion problem.[J].Applied Mathematics and Computation,2003,142(2):417-447. doi: 10.1016/S0096-3003(02)00312-0
[9]	Beckett G, Mackenzie J A. On a uniformly accurate finite difference approximation of a singularly perturbed reaction-diffusion problem using grid equidistribution[J].Journal of Computational and Applied Mathematics,2001,131(1):381-405. doi: 10.1016/S0377-0427(00)00260-0
[10]	Stynes M, Roos H G. The midpoint upwind scheme[J].Appl Numer Math,1997,23(3):361-374. doi: 10.1016/S0168-9274(96)00071-2
[11]	Stynes M, Tobiska L. A finite difference analysis of a streamline diffusion method on a Shishkin mesh[J].Numer Algorithms,1998,18(3):337-360. doi: 10.1023/A:1019185802623
[12]	Rashidiniab J, Ghasemia M, Mahmoodi Z. Spline approach to the solution of a singularly-perturbed boundary value problems[J].Applied Mathematics and Computation,2007,189(1):72-78. doi: 10.1016/j.amc.2006.11.067
[13]	Clavero C, Gracia J. High order methods for elliptic and time dependent reaction-diffusion singularly perturbed problems[J].Applied Mathematics and Computation,2005,169(1):1109-1127.
[14]	Cen Z D. A hybrid difference scheme for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient[J].Applied Mathematics and Computation,2005,169(1):689-699. doi: 10.1016/j.amc.2004.08.051