A meshless structure-preserving quasi-interpolation method for solving Allen–Cahn equations on spheres

IF 4.4 2区数学 Q1 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Mathematics and Computers in Simulation Pub Date : 2025-10-04 DOI:10.1016/j.matcom.2025.09.031

Zhengjie Sun , Yuyan Gao

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引用次数: 0

Abstract

This paper proposes a novel meshless structure-preserving scheme for solving Allen–Cahn equations on spheres. Within the scalar auxiliary variable (SAV) framework, we introduce a kernel-based spherical quasi-interpolation method for spatial discretization on scattered collocation points, complemented by appropriate time integration schemes. Our approach stands out by ensuring unconditional energy stability and overcoming the limitations of traditional mesh-based techniques. Quasi-interpolation methods are known for their simplicity and adaptability to various kernel properties, eliminating the need for large linear systems and making the scheme easy to implement. We provide rigorous theoretical analysis to validate the proposed spherical quasi-interpolation SAV scheme, including well-posedness, energy stability, and error estimates. Numerical examples further demonstrate the convergence of the quasi-interpolation method and the accuracy of simulating Allen–Cahn equations on the sphere.

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求解球面上Allen-Cahn方程的无网格保结构拟插值方法

提出了一种新的求解球面上Allen-Cahn方程的无网格保结构格式。在标量辅助变量（SAV）框架内，我们引入了一种基于核的球面准插值方法，用于分散搭配点的空间离散化，并辅以适当的时间积分方案。我们的方法通过确保无条件的能量稳定性和克服传统基于网格的技术的局限性而脱颖而出。准插值方法以其简单性和对各种核性质的适应性而闻名，消除了对大型线性系统的需要，使方案易于实现。我们提供了严格的理论分析来验证所提出的球面准插值SAV方案，包括适定性，能量稳定性和误差估计。数值算例进一步证明了拟插值方法的收敛性和在球面上模拟Allen-Cahn方程的准确性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematics and Computers in Simulation 数学-计算机：跨学科应用

CiteScore

8.90

自引率

4.30%

发文量

335

审稿时长

54 days

期刊介绍： The aim of the journal is to provide an international forum for the dissemination of up-to-date information in the fields of the mathematics and computers, in particular (but not exclusively) as they apply to the dynamics of systems, their simulation and scientific computation in general. Published material ranges from short, concise research papers to more general tutorial articles. Mathematics and Computers in Simulation, published monthly, is the official organ of IMACS, the International Association for Mathematics and Computers in Simulation (Formerly AICA). This Association, founded in 1955 and legally incorporated in 1956 is a member of FIACC (the Five International Associations Coordinating Committee), together with IFIP, IFAV, IFORS and IMEKO. Topics covered by the journal include mathematical tools in: •The foundations of systems modelling •Numerical analysis and the development of algorithms for simulation They also include considerations about computer hardware for simulation and about special software and compilers. The journal also publishes articles concerned with specific applications of modelling and simulation in science and engineering, with relevant applied mathematics, the general philosophy of systems simulation, and their impact on disciplinary and interdisciplinary research. The journal includes a Book Review section -- and a "News on IMACS" section that contains a Calendar of future Conferences/Events and other information about the Association.