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On the computation of blow-up

Published online by Cambridge University Press:  16 July 2009

A. M. Stuart  and
M. S. Floater
A. M. Stuart
Affiliation:
School of Mathematical Sciences, Bath University, Bath BA2 7AY, UK
M. S. Floater
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK
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Abstract

Numerical methods for initial-value problems which develop singularities in finite time are analyzed. The objective is to determine simple strategies which produce the correct asymptotic behaviour and give an accurate approximation of the blow-up time. Fixed step methods for scalar ordinary differential equations are studied first and it is shown that there is a natural embedding of the discrete process in a continuous one. This shows clearly how and why the fixed-step strategy fails. A class of time-stepping strategies that correspond to a time- continuous re-scaling of the underlying differential equation is then proposed; this class is analyzed and criteria established to determine suitable choices for the re-scaling. Finally the ideas are applied to a partial differential equation arising from the study of a fluid with temperature-dependent viscosity. The numerical method involves re-formulating the equationas a moving boundary problem for the peak value and applying the ODE time-steppingstrategies based on this peak value.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 1 , Issue 1 , March 1990 , pp. 47 - 71
Copyright
Copyright © Cambridge University Press 1990

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