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<p>The convergence of a rescaling algorithm, which is proposed by Berger and Kohn (Commun Pure Appl Math 41:841–863, 1988) to numerically reconstruct the blow-up solutions of nonlinear evolution equations, is analyzed. Berger and Kohn’s algorithm was also used by Anada and Ishiwata (J Differ Equ 262:181–271, 2017) to compute the blow-up rates and was found to be very effective. However, convergence analysis for this algorithm is yet to be studied. In this paper, we consider a typical blow-up problem, the semilinear heat equation, as a model problem for the investigation of the relation between the numerical solution computed by Berger and Kohn’s algorithm and the exact solution. We show the convergence of the computed blow-up time and revisit the blow-up rate in view of Berger and Kohn’s algorithm.</p>

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On the numerical approximation of a rescaling algorithm to nonlinear blow-up problems

  • 期刊论文

  • 发表日期：

    2025年11月24日

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  • Chien-Hong Cho,
  • Hao-Wei Sun

摘要

The convergence of a rescaling algorithm, which is proposed by Berger and Kohn (Commun Pure Appl Math 41:841–863, 1988) to numerically reconstruct the blow-up solutions of nonlinear evolution equations, is analyzed. Berger and Kohn’s algorithm was also used by Anada and Ishiwata (J Differ Equ 262:181–271, 2017) to compute the blow-up rates and was found to be very effective. However, convergence analysis for this algorithm is yet to be studied. In this paper, we consider a typical blow-up problem, the semilinear heat equation, as a model problem for the investigation of the relation between the numerical solution computed by Berger and Kohn’s algorithm and the exact solution. We show the convergence of the computed blow-up time and revisit the blow-up rate in view of Berger and Kohn’s algorithm.