非线性Schrödinger方程的时空有限体积元数值分析

IF 2.3 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-30 DOI:10.1016/j.apnum.2025.10.014

Zhihui Zhao, Hong Li

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引用次数: 0

摘要

本文首次提出了求解三次非线性Schrödinger （NLS）方程的时空有限体积元（STFVE）方法。与时空有限元（STFE）方法相比，该方法不仅易于在空间和时间方向上实现较高的精度，而且该方法本身可以保持物理量的守恒定律，因此很适合求解守恒定律方程。对于所构建的STFVE方案，给出了严格的理论分析，证明了所得到的逼近的存在性，并在空间网格参数与时间步长无关的情况下得到了最优的L∞(L2)和L∞(H1)范数估计。最后，通过数值试验验证了STFVE方法的理论结论、无条件稳定性和守恒性。数值试验表明，该方法能较好地模拟NLS方程。

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Numerical analysis of a space-time finite volume element method for the nonlinear Schrödinger equation

In this paper, for the first time, we propose a space-time finite volume element (STFVE) method for the cubic nonlinear Schrödinger (NLS) equation. In contrast to the space-time finite element (STFE) method, this method not only is easy to achieve high accuracy in both space and time directions, but also the method itself can maintain the conservation laws of physical quantities, and thus it is well suited to solve the conservation laws equations. For the constructed STFVE scheme, the rigorous theoretical analyses are given including the proof of the existence of the resulting approximations and the optimal

L^{\infty} (L^{2})

and

L^{\infty} (H^{1})

norms estimates are obtained in the case that the spatial mesh parameter is not related to the time step size. Finally, some numerical tests are shown to confirm the theoretical findings, unconditional stability and the conservation properties of the STFVE method. Also, the numerical tests show that the STFVE method simulates the NLS equation well.

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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.