三次非线性薛定谔方程的$hr$ -自适应方法
An $hr$-Adaptive Method for the Cubic Nonlinear Schrödinger Equation
摘要：
非线性薛定谔方程（NLSE）是量子力学中最重要的方程之一，在光纤通信、等离子体物理和生物分子动力学等领域有着广泛的应用。这是一个众所周知的困难的问题来解决数值解有非常陡峭的时间和空间梯度。自适应移动网格方法（$r$ -自适应）试图通过将固定数量的节点移动到陡峭解特征区域来优化使用固定数量的节点获得的精度。然而，如果在感兴趣的时间段内解决方案变得或多或少难以解决，则这种方法本身是有限的。网格细化方法（$h$ -自适应），其中网格是局部粗化或细化，是一种替代的自适应策略，这是流行的时间无关的问题。在本文中，我们考虑的组合方法（$hr$ -自适应）的有效性，以解决在一个空间维的NLSE。仿真结果表明，优秀的解决方案的精度相比，其他移动网格的方法。该方法还示出了基于用户的输入误差容限来控制空间误差。证据也表明二阶空间收敛使用一种新的监控功能，以产生自适应移动网格。
英文摘要：
The nonlinear Schrödinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients. Adaptive moving mesh methods ($r$-adaptive) attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features. This approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest. Mesh refinement methods ($h$-adaptive), where the mesh is locally coarsened or refined, is an alternative adaptive strategy which is popular for time-independent problems. In this paper, we consider the effectiveness of a combined method ($hr$-adaptive) to solve the NLSE in one space dimension. Simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches. The method is also shown to control the spatial error based on the user's input error tolerance. Evidence is also presented indicating second-order spatial convergence using a novel monitor function to generate the adaptive moving mesh.
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