The use of positive matrices for the analysis of the large time behavior of the numerical solution of reaction-diffusion systems

Author:
Luciano Galeone

Journal:
Math. Comp. **41** (1983), 461-472

MSC:
Primary 65M10; Secondary 15A51, 65C20

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717696-5

MathSciNet review:
717696

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the numerical solution of nonlinear reaction-diffusion systems with homogeneous Neumann boundary conditions, via the known -method.

We show that if conditions for the positivity of solutions are imposed, then the study of the asymptotic behavior of the numerical solution can be done by means of the theory of stochastic matrices.

In this way it is shown that the numerical solution reproduces the asymptotic behavior of the corresponding theoretical one. In particular, we obtain the decay of the solution to its mean value.

An analysis of the asymptotic stability of the equilibrium points and the convergence of the numerical scheme is given based on the use of *M*-matrices.

Finally we consider the case in which the nonlinear term satisfies a condition of quasimonotonicity.

**[1]**C. Bolley & M. Crouzeix, "Conservation de la positivité lors de la discrétisation des problèmes d'evolution paraboliques,"*RAIRO Anal. Numér.*, v. 12, 1978, pp. 237-245. MR**509974 (80h:35070)****[2]**A. Berman & R. Plemmons,*Nonnegative Matrices in the Mathematical Sciences*, Academic Press, New York, 1979. MR**544666 (82b:15013)****[3]**V. Capasso & S. L. Paveri-Fontana, "Some results on linear stochastic multicompartmental systems,"*Math. Biosci.*, v. 55, 1981, pp. 7-26. MR**625265 (82i:92003)****[4]**S. L. Campbell & C. D. Mayer,*Generalized Inverses of Linear Transformations*, Pitman, London, 1979.**[5]**E. Conway, D. Hoff & J. Smoller, "Large time behavior of solutions of systems of nonlinear reaction-diffusion equations,"*SIAM J. Appl. Math.*, v. 35, 1978, pp. 1-16. MR**0486955 (58:6637)****[6]**P. C. Fife,*Mathematical Aspects of Reacting and Diffusing Systems*, Lecture Notes in Biomath., Vol. 28, Springer-Verlag, New York, 1979. MR**527914 (80g:35001)****[7]**W. E. Fitzgibbon & H. F. Walker,*Nonlinear Diffusion*, Pitman, London, 1977. MR**0442427 (56:809)****[8]**L. Galeone & L. Lopez, "Decay to spatially homogeneous states for the numerical solution of reaction-diffusion systems,"*Calcolo*, v. 19, 1982, pp. 193-208. MR**697462 (84g:65141)****[9]**D. Hoff, "Stability and convergence of finite difference methods for systems of nonlinear reaction-diffusion equations,"*SIAM J. Numer. Anal.*, v. 15, 1978, pp. 1161-1177. MR**512689 (81g:65131)****[10]**J. G. Kemeny & J. L. Snell,*Finite Markov Chains*, Springer-Verlag, New York, 1976. MR**0410929 (53:14670)****[11]**J. P. Lasalle, "Stability theory for difference equations,"*Studies in Ordinary Differential Equations*, Math. Assoc. Amer., 1978. MR**0481689 (58:1789)****[12]**R. H. Martin, "Asymptotic stability and critical points for nonlinear quasimonotone parabolic systems,"*J. Differential Equations*, v. 30, 1978, pp. 301-423. MR**521861 (80g:35011)****[13]**R. D. Richtmyer & K. W. Morton,*Difference Methods for Initial Value Problems*, Interscience, New York, 1967. MR**0220455 (36:3515)****[14]**R. S. Varga,*Matrix Iterative Analysis*, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**0158502 (28:1725)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717696-5

Keywords:
Nonlinear reaction-diffusion systems,
methods,
*M*-matrices,
stochastic matrices,
*A*-stability,
quasimonotone functions

Article copyright:
© Copyright 1983
American Mathematical Society