Analysis of a Crank–Nicolson fast element-free Galerkin method for the nonlinear complex Ginzburg–Landau equation

IF 2.1 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Computational and Applied Mathematics Pub Date : 2024-10-10 DOI:10.1016/j.cam.2024.116323

Xiaolin Li , Xiyong Cui , Shougui Zhang

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引用次数: 0

Abstract

A fast element-free Galerkin (EFG) method is proposed in this paper for solving the nonlinear complex Ginzburg–Landau equation. A second-order accurate time semi-discrete system is presented by using the Crank–Nicolson scheme for the temporal discretization, and then a meshless fully discrete system is established by using the EFG method for the spatial discretization. In the proposed EFG method, Nitsche’s technique is used to impose the essential boundary conditions in a weak sense, and the reproducing kernel gradient smoothing integration is used to accelerate the calculation. Theoretical errors for the time semi-discrete system and the fully discrete EFG system are analyzed in detail. Optimal error estimates of the fully discrete Crank–Nicolson EFG method are obtained in both

L^{2}

and

H^{1}

norms. Numerical results validate the theoretical results and the effectiveness of the method.

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非线性复杂金兹堡-朗道方程的 Crank-Nicolson 快速无元素 Galerkin 方法分析

本文提出了一种用于求解非线性复杂金兹堡-兰道方程的快速无元素伽勒金（EFG）方法。通过使用 Crank-Nicolson 方案进行时间离散化，提出了一个二阶精确时间半离散系统，然后通过使用 EFG 方法进行空间离散化，建立了一个无网格全离散系统。在所提出的 EFG 方法中，使用了 Nitsche 技术来施加弱意义上的基本边界条件，并使用重现核梯度平滑积分来加速计算。详细分析了时间半离散系统和全离散 EFG 系统的理论误差。得到了全离散 Crank-Nicolson EFG 方法在 L2 和 H1 规范下的最佳误差估计值。数值结果验证了理论结果和方法的有效性。

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.