Unconditionally superconvergent error analysis of an energy-conservative Galerkin method for the nonlinear Schrödinger equation with wave operator

IF 3.8 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2026-07-01 Epub Date: 2026-01-29 DOI:10.1016/j.cnsns.2026.109807

Xin Liao, Lele Wang, Huaijun Yang

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Abstract

In this paper, based on the method of order reduction in time, an energy-conservative modified Crank-Nicolson Galerkin scheme is proposed and the unconditionally superconvergent error analysis is investigated for the nonlinear Schrödinger equation with wave operator in two dimensions. The existence and uniqueness of numerical solution are discussed. Unlike the boundedness of numerical solutions in L^∞-norm used in the previous work, the key to our analysis is to novelly employ the boundedness of the numerical solution in H¹-norm derived from the energy-conservative property to deal with the nonlinear term strictly and skillfully. By means of the high accuracy estimate of the bilinear element on the rectangular mesh,the unconditionally superclose error estimate is obtained without any restrictions on the ratio of temporal-spatial step-szie. Furthermore, the unconditionally superconvergence error estimate is acquired by an interpolation post-processing approach. Finally, numerical experiments are carried out to demonstrate the expected accuracy and conservation of proposed schemes.

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带波算子非线性Schrödinger方程的能量守恒Galerkin方法的无条件超收敛误差分析

本文基于时间降阶方法，提出了一种能量保守的修正Crank-Nicolson Galerkin格式，并研究了二维波算子非线性Schrödinger方程的无条件超收敛误差分析。讨论了数值解的存在唯一性。与以往工作中使用的L∞范数数值解的有界性不同，本文分析的关键在于新颖地利用了由能量守恒性质导出的h1 -范数数值解的有界性来严格而巧妙地处理非线性项。通过对矩形网格上双线性单元的高精度估计，在不受时空步长比限制的情况下，获得了无条件超接近误差估计。此外，通过插值后处理方法获得了无条件超收敛误差估计。最后，通过数值实验验证了所提格式的精度和守恒性。

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.