The ADI difference and extrapolation scheme for high-dimensional variable coefficient evolution equations

Jinxiu Zhang; Xuehua Yang; Song Wang; Jinxiu Zhang; Xuehua Yang; Song Wang

doi:10.3934/era.2025146

Electronic Research Archive

2025, Volume 33, Issue 5: 3305-3327. doi: 10.3934/era.2025146

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The ADI difference and extrapolation scheme for high-dimensional variable coefficient evolution equations

1.
School of Science, Hunan University of Technology, Zhuzhou 412007, China
2.
School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, GPO Box U1987, Perth, WA 6845, Australia

Received: 01 April 2025 Revised: 30 April 2025 Accepted: 14 May 2025 Published: 26 May 2025

This paper investigated a numerical method for a two-dimensional variable coefficient evolution equation (VCEE), utilizing the alternating direction implicit (ADI) method and an extrapolation formula. The time derivative was discretised by the backward Euler (BE) scheme on a uniform mesh and the finite difference method (FDM) was applied to spatial discretization. We proved an priori estimate and the error bound of the solution to the difference scheme using the energy analysis method, and verified the uniqueness, stability, and convergence of the proposed scheme. To further improve numerical accuracy, we introduced a Richardson extrapolation method, which enhances the global accuracy to fourth order. Finally, some numerical examples were provided to demonstrate the validity of the theoretical analysis.
- parabolic equation,
- variable coefficients,
- finite difference method,
- alternating direction implicit,
- stability,
- convergence
Citation: Jinxiu Zhang, Xuehua Yang, Song Wang. The ADI difference and extrapolation scheme for high-dimensional variable coefficient evolution equations[J]. Electronic Research Archive, 2025, 33(5): 3305-3327. doi: 10.3934/era.2025146

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Abstract

This paper investigated a numerical method for a two-dimensional variable coefficient evolution equation (VCEE), utilizing the alternating direction implicit (ADI) method and an extrapolation formula. The time derivative was discretised by the backward Euler (BE) scheme on a uniform mesh and the finite difference method (FDM) was applied to spatial discretization. We proved an priori estimate and the error bound of the solution to the difference scheme using the energy analysis method, and verified the uniqueness, stability, and convergence of the proposed scheme. To further improve numerical accuracy, we introduced a Richardson extrapolation method, which enhances the global accuracy to fourth order. Finally, some numerical examples were provided to demonstrate the validity of the theoretical analysis.

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