A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations
Chien-Hong Cho 1*1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan; and Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621, Taiwan.
Received 10 December 2008; Accepted (in revised version) 30 September 2009
We consider a finite difference scheme for a nonlinear wave equation, whose solutions may lose their smoothness in finite time, i.e., blow up in finite time. In order to numerically reproduce blow-up solutions, we propose a rule for a time-stepping, which is a variant of what was successfully used in the case of nonlinear parabolic equations. A numerical blow-up time is defined and is proved to converge, under a certain hypothesis, to the real blow-up time as the grid size tends to zero.AMS subject classifications: 65M06
Key words: Finite difference method, nonlinear wave equation, blow-up.
Email: email@example.com (C.-H. Cho)