On the numerical solutions for a parabolic system with blow-up

Chien-Hong Cho; Ying-Jung Lu; Chien-Hong Cho; Ying-Jung Lu

doi:10.3934/math.2021683

AIMS Mathematics

2021, Volume 6, Issue 11: 11749-11777. doi: 10.3934/math.2021683

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On the numerical solutions for a parabolic system with blow-up

Chien-Hong Cho ^{1
,
,},
Ying-Jung Lu ²

1.
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan
2.
Department of Data Application, USI Corporation, Neihu, Taipei 114, Taiwan

Received: 14 January 2021 Accepted: 08 May 2021 Published: 12 August 2021
MSC : 65M12

We study the finite difference approximation for axisymmetric solutions of a parabolic system with blow-up. A scheme with adaptive temporal increments is commonly used to compute an approximate blow-up time. There are, however, some limitations to reproduce the blow-up behaviors for such schemes. We thus use an algorithm, in which uniform temporal grids are used, for the computation of the blow-up time and blow-up behaviors. In addition to the convergence of the numerical blow-up time, we also study various blow-up behaviors numerically, including the blow-up set, blow-up rate and blow-up in $ L^\sigma $-norm. Moreover, the relation between blow-up of the exact solution and that of the numerical solution is also analyzed and discussed.
- parabolic system,
- blow-up,
- finite difference method,
- blow-up behavior,
- numerical blow-up
Citation: Chien-Hong Cho, Ying-Jung Lu. On the numerical solutions for a parabolic system with blow-up[J]. AIMS Mathematics, 2021, 6(11): 11749-11777. doi: 10.3934/math.2021683

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Abstract

We study the finite difference approximation for axisymmetric solutions of a parabolic system with blow-up. A scheme with adaptive temporal increments is commonly used to compute an approximate blow-up time. There are, however, some limitations to reproduce the blow-up behaviors for such schemes. We thus use an algorithm, in which uniform temporal grids are used, for the computation of the blow-up time and blow-up behaviors. In addition to the convergence of the numerical blow-up time, we also study various blow-up behaviors numerically, including the blow-up set, blow-up rate and blow-up in $ L^\sigma $-norm. Moreover, the relation between blow-up of the exact solution and that of the numerical solution is also analyzed and discussed.

References

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