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Optimal Error Bounds on the Exponential Wave Integrator for the Nonlinear Schrödinger Equation with Low Regularity Potential and Nonlinearity
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2024-01-11 , DOI: 10.1137/23m155414x Weizhu Bao 1 , Chushan Wang 1
SIAM Journal on Numerical Analysis ( IF 2.9 ) Pub Date : 2024-01-11 , DOI: 10.1137/23m155414x Weizhu Bao 1 , Chushan Wang 1
Affiliation
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 93-118, February 2024.
Abstract. We establish optimal error bounds for the exponential wave integrator (EWI) applied to the nonlinear Schrödinger equation (NLSE) with [math]-potential and/or locally Lipschitz nonlinearity under the assumption of [math]-solution of the NLSE. For the semidiscretization in time by the first-order Gautschi-type EWI, we prove an optimal [math]-error bound at [math] with [math] being the time step size, together with a uniform [math]-bound of the numerical solution. For the full-discretization scheme obtained by using the Fourier spectral method in space, we prove an optimal [math]-error bound at [math] without any coupling condition between [math] and [math], where [math] is the mesh size. In addition, for [math]-potential and a little stronger regularity of the nonlinearity, under the assumption of [math]-solution, we obtain an optimal [math]-error bound. Furthermore, when the potential is of low regularity but the nonlinearity is sufficiently smooth, we propose an extended Fourier pseudospectral method which has the same error bound as the Fourier spectral method, while its computational cost is similar to the standard Fourier pseudospectral method. Our new error bounds greatly improve the existing results for the NLSE with low regularity potential and/or nonlinearity. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.
中文翻译：
低正则势非线性薛定谔方程指数波积分器的最优误差界
SIAM 数值分析杂志，第 62 卷，第 1 期，第 93-118 页，2024 年 2 月。
摘要。我们建立了指数波积分器 (EWI) 的最佳误差界，该指数波积分器 (EWI) 应用于非线性薛定谔方程 (NLSE)，在 NLSE 的 [数学] 解的假设下，具有 [数学] 势和/或局部 Lipschitz 非线性。对于一阶 Gautschi 型 EWI 的时间半离散化，我们证明了在 [math] 处的最佳 [math] 误差界，其中 [math] 为时间步长，以及统一的 [math] 界数值解。对于在空间中使用傅里叶谱方法获得的全离散化方案，我们证明了在[math]处的最佳[math]误差界，而[math]和[math]之间没有任何耦合条件，其中[math]是网格尺寸。此外，对于[math]-势和稍强的非线性规律性，在[math]-解的假设下，我们获得了最优的[math]-误差界。此外，当势的规律性较低但非线性足够平滑时，我们提出了一种扩展的傅里叶伪谱方法，其与傅里叶谱方法具有相同的误差界，而其计算成本与标准傅里叶伪谱方法相似。我们的新误差界限极大地改善了具有低规律性潜力和/或非线性的 NLSE 的现有结果。报告了广泛的数值结果，以证实我们的错误估计并证明它们是敏锐的。
更新日期：2024-01-12
Abstract. We establish optimal error bounds for the exponential wave integrator (EWI) applied to the nonlinear Schrödinger equation (NLSE) with [math]-potential and/or locally Lipschitz nonlinearity under the assumption of [math]-solution of the NLSE. For the semidiscretization in time by the first-order Gautschi-type EWI, we prove an optimal [math]-error bound at [math] with [math] being the time step size, together with a uniform [math]-bound of the numerical solution. For the full-discretization scheme obtained by using the Fourier spectral method in space, we prove an optimal [math]-error bound at [math] without any coupling condition between [math] and [math], where [math] is the mesh size. In addition, for [math]-potential and a little stronger regularity of the nonlinearity, under the assumption of [math]-solution, we obtain an optimal [math]-error bound. Furthermore, when the potential is of low regularity but the nonlinearity is sufficiently smooth, we propose an extended Fourier pseudospectral method which has the same error bound as the Fourier spectral method, while its computational cost is similar to the standard Fourier pseudospectral method. Our new error bounds greatly improve the existing results for the NLSE with low regularity potential and/or nonlinearity. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.
中文翻译：
低正则势非线性薛定谔方程指数波积分器的最优误差界
SIAM 数值分析杂志，第 62 卷，第 1 期，第 93-118 页，2024 年 2 月。
摘要。我们建立了指数波积分器 (EWI) 的最佳误差界，该指数波积分器 (EWI) 应用于非线性薛定谔方程 (NLSE)，在 NLSE 的 [数学] 解的假设下，具有 [数学] 势和/或局部 Lipschitz 非线性。对于一阶 Gautschi 型 EWI 的时间半离散化，我们证明了在 [math] 处的最佳 [math] 误差界，其中 [math] 为时间步长，以及统一的 [math] 界数值解。对于在空间中使用傅里叶谱方法获得的全离散化方案，我们证明了在[math]处的最佳[math]误差界，而[math]和[math]之间没有任何耦合条件，其中[math]是网格尺寸。此外，对于[math]-势和稍强的非线性规律性，在[math]-解的假设下，我们获得了最优的[math]-误差界。此外，当势的规律性较低但非线性足够平滑时，我们提出了一种扩展的傅里叶伪谱方法，其与傅里叶谱方法具有相同的误差界，而其计算成本与标准傅里叶伪谱方法相似。我们的新误差界限极大地改善了具有低规律性潜力和/或非线性的 NLSE 的现有结果。报告了广泛的数值结果，以证实我们的错误估计并证明它们是敏锐的。
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