Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Numerical Methods for Partial Differential Equations Pub Date : 2024-08-02 DOI:10.1002/num.23135

Yongyong Cai, Yue Feng, Yichen Guo, Zhiguo Xu

引用次数: 0

Abstract

We propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.

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带波算子的非线性薛定谔方程嵌套皮卡尔积分器的均匀和最优误差估计

我们提出了一种二阶嵌套皮卡尔迭代积分器正弦伪谱（NPI-SP）方法，用于涉及一个参数的带波算子的非线性薛定谔方程（NLSW），并进行了严格的误差估计。实际上，该方程在时间上以波长传播波，而前导振荡的振幅则分别针对准备充分的初始数据和准备不足的初始数据。在指数积分器和嵌套 Picard 迭代的基础上，提出了均匀精度（w.r.t. ）的 NPI-SP 方案，该方案在时间上具有最优均匀误差边界，在空间上对准备充分和准备不足的数据都具有-正态的光谱精度。这一结果极大地改进了有限差分法和指数波积分器对 NLSW 的误差约束。对误差进行了严格估计，并提供了数值示例来证实理论分析。

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

Numerical Methods for Partial Differential Equations 数学-应用数学

CiteScore

7.20

自引率

2.60%

发文量

审稿时长

9 months

期刊介绍： An international journal that aims to cover research into the development and analysis of new methods for the numerical solution of partial differential equations, it is intended that it be readily readable by and directed to a broad spectrum of researchers into numerical methods for partial differential equations throughout science and engineering. The numerical methods and techniques themselves are emphasized rather than the specific applications. The Journal seeks to be interdisciplinary, while retaining the common thread of applied numerical analysis.