An explicit fourth-order accurate compact method for the Allen-Cahn equation

Chaeyoung Lee; Seokjun Ham; Youngjin Hwang; Soobin Kwak; Junseok Kim; Chaeyoung Lee; Seokjun Ham; Youngjin Hwang; Soobin Kwak; Junseok Kim

doi:10.3934/math.2024038

AIMS Mathematics

2024, Volume 9, Issue 1: 735-762. doi: 10.3934/math.2024038

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Research article

An explicit fourth-order accurate compact method for the Allen-Cahn equation

1.
Department of Mathematics, Kyonggi University, Suwon 16227, Republic of Korea
2.
Department of Mathematics, Korea University, Seoul, 02841, Republic of Korea

Received: 20 September 2023 Revised: 21 November 2023 Accepted: 27 November 2023 Published: 04 December 2023
MSC : 65D25, 65N06

In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.
- fourth-order accurate compact method,
- Allen-Cahn equation,
- explicit Euler method
Citation: Chaeyoung Lee, Seokjun Ham, Youngjin Hwang, Soobin Kwak, Junseok Kim. An explicit fourth-order accurate compact method for the Allen-Cahn equation[J]. AIMS Mathematics, 2024, 9(1): 735-762. doi: 10.3934/math.2024038

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Abstract

In this paper, we propose an explicit spatially fourth-order accurate compact scheme for the Allen-Cahn equation in one-, two-, and three-dimensional spaces. The proposed method is based on the explicit Euler time integration scheme and fourth-order compact finite difference method. The proposed numerical solution algorithm is highly efficient and simple to implement because it is an explicit scheme. There is no need to solve implicitly a system of discrete equations as in the case of implicit numerical schemes. Furthermore, when we consider the temporally accurate numerical solutions, the time step restriction is not severe because the governing equation is a second-order parabolic partial differential equation. Computational tests are conducted to demonstrate the superior performance of the proposed spatially fourth-order accurate compact method for the Allen-Cahn equation.

References

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