Communications in Mathematical Sciences

Volume 17 (2019)

Number 4

An application of FDM by Gibou to a numerical blow-up for nonlinear evolution equations on a domain $\Omega \subset \mathbb{R}^N$

Pages: 1149 – 1165

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n4.a15

Authors

Soon-Yeong Chung (Department of Mathematics and Program of Integrated Biotechnology, Sogang University, Seoul, South Korea)

Jea-Hyun Park (Department of Mathematics, Kunsan National University, Kunsan, South Korea)

Abstract

In this paper, using a finite difference method introduced by Gibou et al., we show the blow-up phenomenon of solutions to nonlinear evolution equations with Dirichlet boundary condition on an $N$-dimensional smooth bounded domain $\Omega \subset \mathbb{R}^N$.

We first present bounds of the discrete smallest eigenvalue and the corresponding eigenfunction to the discrete Dirichlet eigenvalue problem with the discrete Laplacian which is obtained by Gibou’s method. We also show that the numerical solution is second order accurate to the theoretical solution and there exists a blow-up time of the numerical solution by finding upper and lower bounds of the blow-up time. Finally, using the above results, we prove that the theoretical solution has a blow-up time and we also give upper and lower bounds for the blow-up time of the theoretical solution.

Keywords

blow-up, nonlinear evolution equations, semidiscretization, convergence analysis

2010 Mathematics Subject Classification

35K61, 65M06

Full Text (PDF format)

Received 23 November 2016

Accepted 4 April 2019

Published 25 October 2019