A decoupled linear, mass- and energy-conserving relaxation-type high-order compact finite difference scheme for the nonlinear Schrödinger equation

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2025-04-15 DOI:10.1016/j.apnum.2025.04.005

Wenrong Zhou , Hongfei Fu , Shusen Xie

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引用次数: 0

Abstract

In this paper, a relaxation-type high-order compact finite difference (RCFD) scheme is proposed for the one-dimensional nonlinear Schrödinger equation. More specifically, the relaxation approach combined with the Crank-Nicolson formula is utilized for time discretization, and fourth-order compact difference method is applied for space discretization. The scheme is linear, decoupled, and can be solved sequentially with respect to the primal and relaxation variables, which avoids solving large-scale nonlinear algebraic systems resulting in fully implicit numerical schemes. Furthermore, the developed scheme is shown to preserve both mass and energy at the discrete level. Most importantly, with the help of a discrete elliptic projection and a cut-off numerical technique, the existence and uniqueness of the high-order RCFD scheme are ensured, and unconditional optimal-order error estimate in discrete maximum-norm is rigorously established. Finally, several numerical experiments are given to support the theoretical findings, and comparisons with other methods are also presented to show the efficiency and effectiveness of our method.

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非线性Schrödinger方程的解耦线性、质量和能量守恒松弛型高阶紧致有限差分格式

本文针对一维非线性Schrödinger方程，提出了一种松弛型高阶紧致有限差分格式。具体而言，时间离散采用松弛法结合Crank-Nicolson公式，空间离散采用四阶紧致差分法。该格式是线性的，解耦的，并且可以根据原始变量和松弛变量顺序求解，从而避免了求解大规模非线性代数系统导致的完全隐式数值格式。此外，所开发的方案被证明在离散水平上同时保持质量和能量。最重要的是，借助离散椭圆投影和截止数值技术，保证了高阶RCFD格式的存在唯一性，并严格建立了离散最大范数下的无条件最优阶误差估计。最后，通过数值实验对理论结果进行了验证，并与其他方法进行了比较，验证了本文方法的有效性。

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

Applied Numerical Mathematics 数学-应用数学

CiteScore

5.60

自引率

7.10%

发文量

225

审稿时长

7.2 months

期刊介绍： The purpose of the journal is to provide a forum for the publication of high quality research and tutorial papers in computational mathematics. In addition to the traditional issues and problems in numerical analysis, the journal also publishes papers describing relevant applications in such fields as physics, fluid dynamics, engineering and other branches of applied science with a computational mathematics component. The journal strives to be flexible in the type of papers it publishes and their format. Equally desirable are: (i) Full papers, which should be complete and relatively self-contained original contributions with an introduction that can be understood by the broad computational mathematics community. Both rigorous and heuristic styles are acceptable. Of particular interest are papers about new areas of research, in which other than strictly mathematical arguments may be important in establishing a basis for further developments. (ii) Tutorial review papers, covering some of the important issues in Numerical Mathematics, Scientific Computing and their Applications. The journal will occasionally publish contributions which are larger than the usual format for regular papers. (iii) Short notes, which present specific new results and techniques in a brief communication.