Blow-Up of Finite-Difference Solutions to Nonlinear Wave Equations

J. Math. Sci. Univ. Tokyo
Vol. 23 (2016), No. 1, Page 349–380.

Saito, Norikazu ; Sasaki, Takiko
Blow-Up of Finite-Difference Solutions to Nonlinear Wave Equations
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Finite-difference schemes for computing blow-up solutions of one dimensional nonlinear wave equations are presented. By applying time increments control technique, we can introduce a numerical blow-up time which is an approximation of the exact blow-up time of the nonlinear wave equation. After having verified the convergence of our proposed schemes, we prove that solutions of those finite-difference schemes actually blow up in the corresponding numerical blow-up times. Then, we prove that the numerical blow-up time converges to the exact blow-up time as the discretization parameters tend to zero. Several numerical examples that confirm the validity of our theoretical results are also offered.

Keywords: Nonlinear wave equation, blow-up, finite-difference method.

Mathematics Subject Classification (2010): 35L71, 35B44, 65M06.
Mathematical Reviews Number: MR3469479

Received: 2014-07-22