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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:[email protected] (C.-H. Cho)
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Abstract

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
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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References

[1] Abia, L., López-Marcos, J.C., and J. Martínez, The Euler method in the numerical integration of reaction-diffusion problems with blow-up, Appl. Numer. Math. 38, 287–313 (2001).Google Scholar
[2] Bizoń, P., Chmaj, T., and Szpak, N., Dynamics near the threshold for blowup in the one-dimensional focusing nonlinear Klein-Gordon equation, J. Math. Phys. 52, 103703–11 (2011).Google Scholar
[3] Brandle, C. and Brunner, H., Blow-up in diffusion equations: a survey, J. Comput. Appl. Math. 97, 3–22 (1998).Google Scholar
[4] Chen, Y.-G., Asymptotic behaviours of blowing-up solutions for finite difference analogue of ut = uxx + u 1+α, J. Fac. Sci. Univ. Tokyo 33, 541–574 (1986).Google Scholar
[5] Cho, C.-H., A finite difference scheme for blow-up solutions of nonlinear wave equations, Numer. Math.: Theo. Methods Appl. 3, 475–498 (2010).Google Scholar
[6] Cho, C.-H., On the convergence of numerical blow-up time for a second order nonlinear ordinary equation, Appl. Math. Lett. 24, 49–54 (2011).Google Scholar
[7] Cho, C.-H., Stability for the finite difference schemes of the linear wave equation with non-uniform time meshes, Numer. Method Partial Differe. Equat. 29, 1031–1042 (2013).Google Scholar
[8] Cho, C.-H., Hamada, S., and Okamoto, H., On the finite differnece approximation for a parabolic blow-up problem, Japan J. Indus. Appl. Math. 24, 131–160 (2007).Google Scholar
[9] Friedman, A. and McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34, 425–447 (1985).Google Scholar
[10] Glassey, R. T., Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177, 323–340 (1981).Google Scholar
[11] Groisman, P., Totally discrete explicit and semi-implicit Euler methods for a blow-up problem in several space dimensions, Computing 76, 325–352 (2006).Google Scholar
[12] John, F., Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28, 235–268 (1979).Google Scholar
[13] Kato, T., Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math. 32, 501–505 (1980).Google Scholar
[14] Levine, H., Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = −Au + F(u), Trans. Amer. Math. Soc. 192, 1–21 (1974).Google Scholar
[15] Killip, R., Stovall, B. and Visan, M., Blowup behavior for the nonlinear Klein-Gordon equation, Math. Ann. 358, 289–350 (2014).Google Scholar
[16] Matus, Piotr P., Mazhukin, Vladimir I., and Mozolevsky, Igor E., Stability of finite difference schemes on non-uniform spatial-time-grids, NAA 2000, LNCS 1988, 568–577 (2001).Google Scholar
[17] Matus, P. and Zyuzina, E., Three-level difference schemes on non-uniform in time grids, Comput. Methods Appl. Math. 1, 265–284 (2001).Google Scholar
[18] Nakagawa, T., Blowing up of a finite difference solution to ut = uxx +u 2, Appl. Math. Optim. 2, 337–350 (1976).Google Scholar
[19] Saito, N. and Sasaki, T., Blow-up of finite-difference solutions to nonlinear wave equations, J. Math. Sci. Univ. Tokyo 23, 349–380 (2016).Google Scholar
[20] Samarskii, A. A., Vabishchevich, P. N., Makarevich, E. L., and Matus, P. P., Stability of three-layer difference schemes on time-nonuniform grids, Dokl. Russ. Acad. Nauk 376, 738–741 (2001).Google Scholar