Point-wise error estimate of a conservative difference scheme for the fractional Schrodinger equation
Point-wise error estimate of a conservative difference scheme for the fractional Schrodinger equation
复制标题
分数阶薛定谔方程保守差分格式的逐点误差估计
DOI：
10.1016/j.cam.2016.04.017
复制
发表时间：
2016
影响因子：
2.4
通讯作者：
Zhao Longbin
中科院分区：
文献类型：
--
作者：
Wang Pengde;Huang Chengming;Zhao Longbin
本文提出了一种新的求解非线性空间分数阶薛定谔方程的守恒差分格式。Riesz空间分数阶导数由二阶精确的加权移位Grünwald差分算子近似。在此基础上，分析了该算子的质量和能量守恒性质，建立了一些范数等价关系。然后证明了数值解关于离散参数一致地在各种范数下有界，并给出了格式的全局误差在l2范数、半H α/2范数和l∞范数下的最优阶界.数值解的统一边界以及误差边界无条件地保持，在这个意义上，没有限制的空间网格尺寸方面的时间步长的大小需要被假定。数值试验支持我们的理论结果。
In this paper, a new conservative difference scheme is proposed for solving the nonlinear space-fractional Schrödinger equation. The Riesz space-fractional derivative is approximated by the second-order accurate weighted and shifted Grünwald difference operator. Building on a careful analysis of this operator, the conservation properties including the mass and energy are analyzed and some norm equivalences are established. Then the numerical solution is shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme in the l 2 norm, semi-H α/2 norm and l∞ norm are derived. The uniform bounds on the numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed. Numerical tests are performed to support our theoretical results.