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Numer. Math. Theor. Meth. Appl., 3 (2010), pp. 475-498. |
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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 Abstract 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: 65M06Key words: Finite difference method, nonlinear wave equation, blow-up. *Corresponding author. Email: cho1003jp@yahoo.co.jp (C.-H. Cho) |