Stability analysis of a class of nonlinear magnetic diffusion equations and its fully implicit scheme

Gao Chang; Chunsheng Feng; Jianmeng He; Shi Shu; Gao Chang; Chunsheng Feng; Jianmeng He; Shi Shu

doi:10.3934/math.20241014

AIMS Mathematics

2024, Volume 9, Issue 8: 20843-20864. doi: 10.3934/math.20241014

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Stability analysis of a class of nonlinear magnetic diffusion equations and its fully implicit scheme

1.
School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 411105, China
2.
College of General Education, Shanxi Institute of Science and Technology, Jincheng, Shanxi, 048000, China

Received: 06 May 2024 Revised: 06 June 2024 Accepted: 13 June 2024 Published: 27 June 2024
MSC : 35L03, 35L65, 65M08

We studied a class of nonlinear magnetic diffusion problems with step-function resistivity $ \eta(e) $ in electromagnetically driven high-energy-density physics experiments. The stability of the nonlinear magnetic diffusion equation and its fully implicit scheme, based on the step-function resistivity approximation model $ \eta_\delta(e) $ with smoothing, were studied. A rigorous theoretical analysis was established for the approximate model of one-dimensional continuous equations using Gronwall's theorem. Following this, the stability of the fully implicit scheme was proved using bootstrapping and other methods. The correctness of the theoretical proof was verified through one-dimensional numerical experiments.
- nonlinear magnetic diffusion equation,
- step-function resistivity,
- stability,
- implicit finite volume method
Citation: Gao Chang, Chunsheng Feng, Jianmeng He, Shi Shu. Stability analysis of a class of nonlinear magnetic diffusion equations and its fully implicit scheme[J]. AIMS Mathematics, 2024, 9(8): 20843-20864. doi: 10.3934/math.20241014

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Abstract

We studied a class of nonlinear magnetic diffusion problems with step-function resistivity $ \eta(e) $ in electromagnetically driven high-energy-density physics experiments. The stability of the nonlinear magnetic diffusion equation and its fully implicit scheme, based on the step-function resistivity approximation model $ \eta_\delta(e) $ with smoothing, were studied. A rigorous theoretical analysis was established for the approximate model of one-dimensional continuous equations using Gronwall's theorem. Following this, the stability of the fully implicit scheme was proved using bootstrapping and other methods. The correctness of the theoretical proof was verified through one-dimensional numerical experiments.

References

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