Numerical Solution of a nonlinear reaction-diffusion problem in the case of HS-regime

TL;DR

This paper introduces a numerical approach for solving a nonlinear reaction-diffusion problem in the HS-regime, combining hyperbolic and parabolic methods, and demonstrates finite-time blow-up of solutions.

Contribution

The paper presents a novel numerical scheme that integrates hyperbolic and parabolic techniques for HS-regime reaction-diffusion problems, with theoretical blow-up analysis.

Findings

Numerical solutions blow up in finite time.

The method effectively combines Hopf-Lax and finite element techniques.

Estimates of the numerical solution are established.

Abstract

In this paper, the authors propose a numerical method to compute the solution of a nonlinear reaction-diffusion problem in the case of HS-regime. The initial condition is a nonnegative function with compact support. The problem is split in two parts: A hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a finite element method in space. Estimates of the numerical solution are obtained and it is proved that any numerical solution blows up in finite time.