Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-26T17:27:35.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence and stability analysis of an explicit finite difference method for 2-dimensional reaction-diffusion equations

Published online by Cambridge University Press:  17 February 2009

Nian Li ,
Joseph Steiner  and
Shimin Tang
Nian Li
Affiliation:
Mathematics Department, Swinburne University of Technology, Hawthorn 3122, Australia.
Joseph Steiner
Affiliation:
Mathematics Department, Swinburne University of Technology, Hawthorn 3122, Australia.
Shimin Tang
Affiliation:
Department of Mechanics, Peking University, Beijing, China.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The convergence and stability analysis of a simple explicit finite difference method is studied in this paper. Conditional convergence and stability theorems for this method are given. We have also proved that this scheme is stable in a much stronger sense.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 2 , October 1994 , pp. 234 - 241
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Argyris, J., Haase, M. and Heinrich, J. C., “Finite approximation to two-dimensional sin-Gordon equations”, Comput. Methods Appl. Mech. Engrg. 86 (1991) 1–26.CrossRefGoogle Scholar
[2]Aronson, D. G. and Weinberger, H. F., “Multidimensional nonlinear diffusion arising in population genetics”, Adv. in Math. 30 (1978) 33–76.CrossRefGoogle Scholar
[3]Evans, D. J. and Sahimi, M. S., “The alternating group explicit (AGE) iterative method to solve parabolic and hyperbolic partial differential equations”, Ann. of Numerical Fluid Mechanics and Heat Transfer 2 (1989) 283–389.Google Scholar
[4]Gazdag, J. and Canosa, J., “Numerical solutions of Fisher's equation”, J. Appl. Probab. 11 (1974) 445–457.CrossRefGoogle Scholar
[5]Godunov, S. K. and Ryabenkii, V. S., Difference schemes (Elsevier Science, New York, 1987).Google Scholar
[6]Golub, G. H. and Loan, C. F. Van, Matrix computations (John Hopkins, Baltimore, 1989).Google Scholar
[7]Griffiths, D. F. and Mitchell, A. R., “Stable periodic bifurcations of an explicit discretization of a nonlinear partial differential equation in reaction diffusion”, IMA J. Numer. Anal. 8 (1988) 435–454.CrossRefGoogle Scholar
[8]Grimshaw, R. and Tang, S., “The rotation-modified Kadomtsev-Petviashvili equation: an analytical and numerical study”, Stud. Appl. Math. 83 (1990) 223–248.CrossRefGoogle Scholar
[9]Hoff, D., “Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations”, SIAM J. Numer. Anal. 15 (6) (1978) 1161–1177.CrossRefGoogle Scholar
[10]Isaacson, E. and Keller, H. B., Analysis of numerical methods (Wiley, New York, 1966).Google Scholar
[11]Marchuk, G. I., Methods of numerical mathematics (Springer-Verlag, New York, 1975).Google Scholar
[12]O'Brien, G. G., Hyman, M. A. and Kaplan, S., “A study of the numerical solution of partial differential equation”, J. Math. Phys. 29 (1951) 223.CrossRefGoogle Scholar
[13]Richtmeyer, R. and Morton, K. W., Difference methods for initial value problems (Wiley, New York, 1967).Google Scholar
[14]Tang, S., Qin, S. and Weber, R. O., “Numerical solution of a nonlinear reaction-diffusion equation”, Applied mathematics and mechanics, English edition 12 (8) (08 1991) 751–758.Google Scholar
[15]Tang, S., Qin, S. and Weber, R. O., “Numerical studies on 2-dimensional reaction-diffusion equations”, J. Austral. Math. Soc. Ser. B 35 (1993) 223–243.CrossRefGoogle Scholar
[16]Neumann, J. von and Richtmeyer, R. D., “A method for the numerical calculation of hydrodynamical shocks”, J. Appl. Phys. 21 (1950) 232.CrossRefGoogle Scholar
[17]Yu, G. B. and Mitchell, A. R., “Analysis of a non-linear difference scheme in reaction-diffusion”, Numer. Math. 49 (1986) 511–527.CrossRefGoogle Scholar