The nonlinear Schr\"{o}dinger (NLS) equation possesses an infinite hierarchy of conserved densities and the numerical preservation of some of these quantities is critical for accurate long-time simulations, particularly for multi-soliton solutions. We propose an essentially explicit discretization that conserves one or two of these conserved quantities by combining higher-order Implicit-Explicit (ImEx) Runge-Kutta time integrators with the relaxation technique and adaptive step size control. We show through numerical tests that our mass-conserving method is much more efficient and accurate than the widely-used 2nd-order time-splitting pseudospectral approach. Compared to higher-order operator splitting, it gives similar results in general and significantly better results near the semi-classical limit. Furthermore, for some problems adaptive time stepping provides a dramatic reduction in cost without sacrificing accuracy. We also propose a full discretization that conserves both mass and energy by using a conservative finite element spatial discretization and multiple relaxation in time. Our results suggest that this method provides a qualitative improvement in long-time error growth for multi-soliton solutions.

翻译：非线性薛定谔方程具有无限层次守恒密度，对这些量的数值保守恒对于长时间高精度模拟——尤其是多孤子解问题——至关重要。本文提出一种本质上显式的离散格式，通过将高阶隐式-显式龙格-库塔时间积分器与松弛技术及自适应步长控制相结合，实现对一至两个守恒量的保守恒。数值测试表明，本文提出的质量守恒方法相比广泛使用的二阶时间分裂拟谱方法具有更高的效率和精度。与高阶算子分裂方法相比，该方法在一般情况下可获得相似结果，而在半经典极限附近则表现出显著优势。此外，对于特定问题，自适应时间步长可在不牺牲精度的前提下大幅降低计算成本。本文还提出通过采用守恒型有限元空间离散与多重时间松弛策略，实现质量和能量同时守恒的全离散格式。结果表明，该方法在多孤子解问题的长时间误差增长特性上具有定性改善。