Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes

✍ Scribed by Alain-Yves Le Roux; Marie-Noelle Le Roux

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 339 KB
• Volume: 214
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2007.02.035

✦ Synopsis

In this paper, the authors propose a numerical method to compute the solution of the Cauchy problem: w t -(w m w x ) x = w p , the initial condition is a nonnegative function with compact support, m > 0, p m + 1. The problem is split into 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. The convergence of the scheme is obtained. Further, it is proved that if m + 1 p < m + 3, any numerical solution blows up in a finite time as the exact solution, while for p > m + 3, if the initial condition is sufficiently small, a global numerical solution exists, and if p m + 3, for large initial condition, the solution is unbounded.