Improved Error Bounds of the Strang Splitting Method for the Highly Oscillatory Fractional Nonlinear Schrödinger Equation

2021 ◽  
Vol 88 (2) ◽  
Author(s):  
Yue Feng
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Error Bounds ◽  
Schrodinger Equation ◽  
Splitting Method ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Strang Splitting ◽  
Highly Oscillatory ◽  
Fractional Nonlinear Schrödinger Equation

scholarly journals On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential

2020 ◽  
Vol 54 (5) ◽  
pp. 1491-1508 ◽  
Author(s):  
Chunmei Su ◽  
Xiaofei Zhao
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fixed Time ◽  
Nonlinear Schrodinger Equation ◽  
Splitting Scheme ◽  
Oscillatory Potential ◽  
Nonlinear Schrödinger ◽  
Strang Splitting ◽  
Highly Oscillatory

In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.

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