Adaptive Absorbing Boundary Layer for the Nonlinear Schrödinger Equation

IF 1 4区 数学 Q3 MATHEMATICS, APPLIED Computational Methods in Applied Mathematics Pub Date : 2023-11-01 DOI:10.1515/cmam-2023-0096
Hans Peter Stimming, Xin Wen, Norbert J. Mauser
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Abstract

Abstract We present an adaptive absorbing boundary layer technique for the nonlinear Schrödinger equation that is used in combination with the Time-splitting Fourier spectral method (TSSP) as the discretization for the NLS equations. We propose a new complex absorbing potential (CAP) function based on high order polynomials, with the major improvement that an explicit formula for the coefficients in the potential function is employed for adaptive parameter selection. This formula is obtained by an extension of the analysis in [R. Kosloff and D. Kosloff, Absorbing boundaries for wave propagation problems, J. Comput. Phys. 63 1986, 2, 363–376]. We also show that our imaginary potential function is more efficient than what is used in the literature. Numerical examples show that our ansatz is significantly better than existing approaches. We show that our approach can very accurately compute the solutions of the NLS equations in one dimension, including in the case of multi-dominant wave number solutions.
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非线性Schrödinger方程的自适应吸收边界层
摘要提出了一种非线性Schrödinger方程的自适应吸收边界层技术,该技术与分时傅立叶谱方法(TSSP)相结合,用于NLS方程的离散化。本文提出了一种新的基于高阶多项式的复吸收势函数,主要改进是采用显式公式对势函数中的系数进行自适应参数选择。该公式由[R]中的分析推广而得。波传播问题的吸收边界,J.计算机学报。物理学报,2002,23(2):363-376。我们还证明了我们的虚势函数比文献中使用的函数更有效。数值算例表明,我们的方法明显优于现有的方法。我们表明,我们的方法可以非常准确地计算一维NLS方程的解,包括在多主导波数解的情况下。
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来源期刊
CiteScore
2.40
自引率
7.70%
发文量
54
期刊介绍: The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics. The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.
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