Direct and indirect approximations to positive solution for a nonlinear reaction-diffusion problem. II. Indirect approximation

Călin Ioan Gheorghiu; Damian Trif

Authors

Călin Ioan Gheorghiu Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy
Damian Trif “Babes-Bolyai” University, Cluj-Napoca, Romania

Keywords:

nonlinear reaction-diffusion, positive solution, conserved integral, projection-like method, f.e.m., finite elements-finite differences method, nonlinear stability, energetic method

Abstract

In this paper we continue the study initiated in our previous work [3] and design a projection-like algorithm to approximate a hyperbolic unstable "point''. This "point'' is in fact the positive solution of the reaction-diffusion problem considered in [3] and the algorithm modifies a finite difference (Euler)-finite elements scheme by incorporating the independence of the length of the domain condition. The numerical results are in good agreement with those obtained by direct methods as well as with those reported in [2], where the problem is solved in a Hamiltonian setting. At the same time we improve our previous results reported in [3].

Downloads

Download data is not yet available.

References

Beyn, W.-J., On the numerical approximation of phase portraits near stationary points, SIAM J. Numer. Anal., 24, pp. 1095-1113, 1987, https://doi.org/10.1137/0724072

Gheorghiu, C. I. and Mureşan, A., On the significance of integral properties of orbits in some superlinear fixed-period problems, Proc. ICNODEA, Cluj-Napoca, 2001.

Gheorghiu, C. I. and Trif, D., Direct and indirect approximations to positive solutions for a nonlinear reaction-diffusion problem, I. Direct (variational) approximation, this journal, 31 no. 1, pp. 61-70, 2002, http://ictp.acad.ro/jnaat/journal/article/view/2002-vol31-no1-art8

Gheorghiu, C. I. and Trif, D., The numerical approximation to positive solution for some reaction-diffusion problems, Pu.M.A., 11, pp. 243--253, 2001.

Henry, D., Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, 840, Springer-Verlag, 1981.

Iserles, A., A First Course in the Numerical Analysis of Differential Equations, Cambridge Univ. Press, 1996.

Larsson, S. and Sanz-Serna, J. M., The behaviour of finite element solutions of semilinear parabolic problems near stationary points, SIAM J. Numer. Anal., 31, pp. 1000-1018, 1994, https://doi.org/10.1137/0731053

Stuart, A. M. and Humphries, A. R., Dynamical systems and numerical analysis, Cambridge Monographs on Applied and Computational Mathematics, Cambridge Univ. Press., 1996.