A Cartesian grid-based kernel-free boundary integral method for Allen-Cahn equation

Min Zeng; Yaning Xie; Min Zeng; Yaning Xie

doi:10.3934/era.2025324

Electronic Research Archive

2025, Volume 33, Issue 12: 7331-7359. doi: 10.3934/era.2025324

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A Cartesian grid-based kernel-free boundary integral method for Allen-Cahn equation

Min Zeng ,
Yaning Xie ^,

School of Mathematical Sciences, Zhejiang University of Technology, Hangzhou 310023, China

Received: 07 October 2025 Revised: 18 November 2025 Accepted: 27 November 2025 Published: 04 December 2025

This paper investigates the numerical solution for Allen-Cahn equations with perturbation parameters and strong nonlinear terms in general computational domains. To get a global second-order accuracy, we use the second-order semi-implicit Runge-Kutta (SIRK) method and some implicit-explicit (IMEX) methods for time discretization. Then the original problem is transformed into boundary value problems (BVPs) of a modified Helmholtz equation at each time step, which can be solved by a Cartesian grid-based kernel-free boundary integral (KFBI) method. In the KFBI method, the BVPs are reformulated into a corresponding boundary integral equation and then solved iteratively by a class of subspace methods such as the matrix-free generalized minimal residual(GMRES) method, while integrals involved are regarded as solutions to their equivalent interface problems. Unlike traditional boundary integral methods, this method avoids numerical integration of the singular or nearly singular integrals. Instead, it utilizes grid-based operations as an alternative to direct evaluation. Therefore, integral evaluation only requires solving equivalent but much simpler interface problems in a bounding box so that fast elliptic solvers such as fast fourier transforms(FFTs) and geometric multigrid methods are applicable. This makes the KFBI method accurate and efficient when solving constant coefficient elliptic problems in general irregular domains. It can be seen that the accuracy of the present method is verified by numerical examples.
- Allen-Cahn equation,
- kernel-free boundary integral method,
- implicit-explicit method,
- boundary value problem,
- interface problem
Citation: Min Zeng, Yaning Xie. A Cartesian grid-based kernel-free boundary integral method for Allen-Cahn equation[J]. Electronic Research Archive, 2025, 33(12): 7331-7359. doi: 10.3934/era.2025324

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Abstract

This paper investigates the numerical solution for Allen-Cahn equations with perturbation parameters and strong nonlinear terms in general computational domains. To get a global second-order accuracy, we use the second-order semi-implicit Runge-Kutta (SIRK) method and some implicit-explicit (IMEX) methods for time discretization. Then the original problem is transformed into boundary value problems (BVPs) of a modified Helmholtz equation at each time step, which can be solved by a Cartesian grid-based kernel-free boundary integral (KFBI) method. In the KFBI method, the BVPs are reformulated into a corresponding boundary integral equation and then solved iteratively by a class of subspace methods such as the matrix-free generalized minimal residual(GMRES) method, while integrals involved are regarded as solutions to their equivalent interface problems. Unlike traditional boundary integral methods, this method avoids numerical integration of the singular or nearly singular integrals. Instead, it utilizes grid-based operations as an alternative to direct evaluation. Therefore, integral evaluation only requires solving equivalent but much simpler interface problems in a bounding box so that fast elliptic solvers such as fast fourier transforms(FFTs) and geometric multigrid methods are applicable. This makes the KFBI method accurate and efficient when solving constant coefficient elliptic problems in general irregular domains. It can be seen that the accuracy of the present method is verified by numerical examples.

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