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On the convergence of the numerical blow-up time for a rescaling algorithm

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Abstract

Berger and Kohn (Comm. Pure Appl. Math. 41, 841–863 1988) proposed an algorithm to compute approximate blow-up times for those evolution equations whose solutions blow up in a finite time and possess a scaling invariance property. Later, Anada et. al (Japan J. Indust. Appl. Math. 35, 33–47 2018) used this algorithm to compute the blow-up rates of various blow-up problems which turned out to be very effective. The convergence analysis for this algorithm, however, seems to be not well-studied yet. As a result, we will verify in this paper the convergence of the numerical blow-up time computed by this algorithm via a nonlinear ODE blow-up problem. The convergence order of the numerical blow-up times is also verified. It should be noted that the numerical blow-up time is given by an infinite sum, which cannot be done in real computation. We thus propose a criterion to stop the computation and define the time corresponding to the stopping step to be the modified numerical blow-up time. We prove the convergence of the modified numerical blow-up times and show that the convergence order can be kept the same as that of the original numerical blow-up time.

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Funding

The first author is supported by the grant MOST 111-2115-M-110-002-MY2, Taiwan.

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  1. Department of Applied Mathematics, National Sun Yat-sen University, Lien-hai Road, Kaohsiung, 804201, Taiwan

    Chien-Hong Cho & Jhih-Sin Wu

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  1. Chien-Hong Cho
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Both authors contributed to the theoretical analyses and numerical simulations in the manuscript. C.-H. Cho wrote the main manuscript text. All authors reviewed the final manuscript.

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Correspondence to Chien-Hong Cho.

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Cho, CH., Wu, JS. On the convergence of the numerical blow-up time for a rescaling algorithm. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01791-2

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