SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1901-1928, August 2024.
Abstract. We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form [math] with [math] being the wave function and [math] being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For “good” potential and nonlinearity ([math]-potential and [math]), we establish an optimal second-order error bound in the [math]-norm. For low regularity potential and nonlinearity ([math]-potential and [math]), we obtain a first-order [math]-norm error bound accompanied with a uniform [math]-norm bound of the numerical solution. Moreover, adopting a new technique of regularity compensation oscillation to analyze error cancellation, for some nonresonant time steps, the optimal second-order [math]-norm error bound is proved under a weaker assumption on the nonlinearity: [math]. For all the cases, we also present corresponding fractional order error bounds in the [math]-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.

中文翻译：

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低正则势和非线性非线性薛定谔方程的显式对称指数波积分器


《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1901-1928 页，2024 年 8 月。

抽象的。我们提出并分析了一种新颖的对称高齐型指数波积分器（sEWI），用于非线性薛定谔方程（NLSE），具有低正则势和典型的幂型非线性，形式为[math]，其中[math]是波函数，[ math] 是非线性的指数。 sEWI 在时间步长限制下是明确且稳定的，与网格尺寸无关。我们在电势和非线性的各种规律性假设下严格建立了 sEWI 的误差估计。对于“良好”的势和非线性（[数学]势和[数学]），我们在[数学]范数中建立了最佳二阶误差界。对于低正则势和非线性（[math]-势和[math]），我们获得一阶[math]-范数误差界以及数值解的统一[math]-范数界。此外，采用规律性补偿振荡新技术来分析误差抵消，对于一些非共振时间步长，在较弱的非线性假设下证明了最优二阶范数误差界：[math]。对于所有情况，我们还在[数学]范数中提出了相应的分数阶误差界限，这是能量方面的自然范数。大量的数值结果证实了我们的误差估计，并证明了 sEWI 的优越性，包括对势能和非线性的规律性要求要弱得多，以及质量和能量接近守恒的优异的长期行为。