Error analysis of the explicit-invariant energy quadratization (EIEQ) numerical scheme for solving the Allen–Cahn equation

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics Pub Date : 2024-09-16 DOI:10.1016/j.cam.2024.116224

Jun Zhang , Fangying Song , Xiaofeng Yang , Yu Zhang

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引用次数: 0

Abstract

This paper focuses on the error analysis of a first-order, time-discrete scheme for solving the nonlinear Allen–Cahn equation. The discretization of the nonlinear potential is achieved through the EIEQ method, which employs an auxiliary variable to linearize the nonlinear double-well potential effectively. The energy stability of the scheme is demonstrated, along with its decoupled type implementation. Under a set of reasonable assumptions related to boundedness and continuity, an extensive error analysis is performed. This analysis results in the establishment of

L^{2}

and

H^{1}

error bounds for the numerical solution. Furthermore, a variety of numerical examples are conducted to illustrate the accuracy of the EIEQ scheme, highlighting its effectiveness in addressing complex dynamical systems governed by the Allen–Cahn equation.

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用于求解艾伦-卡恩方程的显式不变能量四分法（EIEQ）数值方案的误差分析

本文重点分析了求解非线性 Allen-Cahn 方程的一阶时间离散方案的误差。非线性势的离散化是通过 EIEQ 方法实现的，该方法采用了一个辅助变量来有效地线性化非线性双阱势。演示了该方案的能量稳定性及其解耦类型的实现。在一系列与有界性和连续性相关的合理假设下，进行了广泛的误差分析。通过分析，建立了数值解的 L2 和 H1 误差边界。此外，还通过各种数值示例说明了 EIEQ 方案的准确性，突出了它在处理受 Allen-Cahn 方程控制的复杂动力系统时的有效性。

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

Journal of Computational and Applied Mathematics 数学-应用数学

CiteScore

5.40

自引率

4.20%

发文量

437

审稿时长

3.0 months

期刊介绍： The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest. The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.