Volume 52, pp. 391-415, 2020.

Finite difference schemes for an axisymmetric nonlinear heat equation with blow-up

Chien-Hong Cho and Hisashi Okamoto


We study finite difference schemes for axisymmetric blow-up solutions of a nonlinear heat equation in higher spatial dimensions. The phenomenology of blow-up in higher-dimensional space is much more complex than that in one space dimension. To obtain a more complete picture for such phenomena from computational results, it is useful to know the technical details of the numerical schemes for higher spatial dimensions. Since first-order differentiation appears in the differential equation, we pay special attention to it. A sufficient condition for stability is derived. In addition to the convergence of the numerical blow-up time, certain blow-up behaviors, such as blow-up sets and blow-up in the $L^p$-norm, are taken into consideration. It is sometimes experienced that a certain property of solutions of a partial differential equation may be lost by a faithfully constructed convergent numerical scheme. The phenomenon of one-point blow-up is a typical example in the numerical analysis of blow-up problems. We prove that our scheme can preserve such a property. It is also remarkable that the $L^p$-norm $(1\leq p<\infty)$ of the solution of the nonlinear heat equation may blow up simultaneously with the $L^\infty$-norm or remains bounded in $[0,T)$, where $T$ denotes the blow-up time of the $L^\infty$-norm. We propose a systematic way to compute numerical evidence of the $L^p$-norm blow-up. The computational results are also analyzed. Moreover, we prove an abstract theorem which shows the relationship between the numerical $L^p$-norm blow-up and the exact $L^p$-norm blow-up. Numerical examples for higher-dimensional blow-up solutions are presented and discussed.

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Key words

blow-up, finite difference method, nonlinear heat equation, $L^p$-norm blow-up

AMS subject classifications

65M06, 65M12

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