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Volume 7, Issue 4
Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-up Problem

Chien-Hong Cho & Chun-Yi Liu

East Asian J. Appl. Math., 7 (2017), pp. 679-696.

Published online: 2018-02

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  • 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.

  • AMS Subject Headings

65L12

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COPYRIGHT: © Global Science Press

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@Article{EAJAM-7-679, author = {Chien-Hong Cho and Chun-Yi Liu}, title = {Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-up Problem}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {4}, pages = {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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.220816.300517a}, url = {http://global-sci.org/intro/article_detail/eajam/10713.html} }
TY - JOUR T1 - Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-up Problem AU - Chien-Hong Cho & Chun-Yi Liu JO - East Asian Journal on Applied Mathematics VL - 4 SP - 679 EP - 696 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.220816.300517a UR - https://global-sci.org/intro/article_detail/eajam/10713.html KW - Blow-up, numerical blow-up time, finite difference method, nonlinear ODE. AB -

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.

Chien-Hong Cho and Chun-Yi Liu. (2018). Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-up Problem. East Asian Journal on Applied Mathematics. 7 (4). 679-696. doi:10.4208/eajam.220816.300517a
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