Application of Topological Techniques to the Analysis of Asymptotic Behavior of Numerical Solutions of a Reaction-Diffusion Equation
Sat Nam S. Khalsa · SIAM Journal on Mathematical Analysis · 1987
The initial boundary value problem for a reaction-diffusion equation \[ (*)\quad u_t = u_{xx} + f(u),\quad f(u) = - u(u - b)(u - 1),\quad 0 < b < \frac{1}{{2}},\] was analysed in [2], [10] by using the Conley index. In this paper we study the asymptotic behavior of solutions of the semidiscrete approximations \[ (**)\quad \dot u_i = {{(u_{i - 1} - 2u_i + u_{i + 1} )} / {h^2 + f(u_i )}},\quad i = 1, \cdots ,n.\] We show that for large n the spectrum of the linearized discrete steady-state problem is a “good” approximation for the spectrum of the linearized continuous steady-state problem. Using the interpretation of the Conley index as the dimension of an unstable manifold of a steady-state solution, we establish that the properties of solutions of (**) are completely analogous to those of the solutions of (*). The asymptotic, as $t \to \infty $, second order convergence of the approximate solutions is proved.