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Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-Up Problem

Part of:Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  31 January 2018

Chien-Hong Cho and
Chun-Yi Liu
Chien-Hong Cho*
Affiliation:
Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621, Taiwan
Chun-Yi Liu
Affiliation:
Department of Mathematics, National Chung Cheng University, Min-Hsiung, Chia-Yi 621, Taiwan
*
*Corresponding author. Email address:chcho20@ccu.edu.tw (C.-H. Cho)

Abstract

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We consider the second order nonlinear ordinary differential equation u″ (t) = u1+α (α > 0) with positive initial data u(0) = a0, u′(0) = a1, 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.

Keywords

Blow-upnumerical blow-up timefinite difference methodnonlinear ODE

MSC classification

Secondary:65L12: Finite difference methods

Information

Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017, pp. 679 - 696
Copyright
Copyright © Global-Science Press 2017 

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