Numerical solution of a nonlinear reaction diffusion equation

Abstract

In this paper, the authors propose a numerical method to compute the solution of a Cauchy problem with blow-up of the solution. The problem is split in two parts: a hyperbolic problem which is solved by using Hopf and Lax formula and a parabolic problem solved by a backward linearized Euler method in time and a finite element method in space. It is proved that the numerical solution blows up in a finite time as the exact solution and the support of the approximation of a self-similar solution remains bounded. The convergence of the scheme is obtained.