Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-up Problem
Year: 2017
East Asian Journal on Applied Mathematics, Vol. 7 (2017), Iss. 4 : pp. 679–696
Abstract
We consider the second order nonlinear ordinary differential equation $u′′(t)=u^{1+α}(α>0)$ with positive initial data $u(0)=a_0$ , $u′(0)=a_1$ , whose solution becomes unbounded in a finite time $T$. The finite time $T$ is called the blow-up time. Since finite difference schemes with uniform meshes can not reproduce such a phenomenon well, adaptively-defined grids are applied. Convergence with mesh sizes of certain smallness has been considered before. However, more iterations are required to obtain an approximate blow-up time if smaller meshes are applied. As a consequence, we consider in this paper a finite difference scheme with a rather larger grid size and show the convergence of the numerical solution and the numerical blow-up time. Application to the nonlinear wave equation is also discussed.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/eajam.220816.300517a
East Asian Journal on Applied Mathematics, Vol. 7 (2017), Iss. 4 : pp. 679–696
Published online: 2017-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18
Keywords: Blow-up numerical blow-up time finite difference method nonlinear ODE.