Open Access

ARTICLE

Computation of Nonlinear Schrödinger Equation on an Open Waveguide Terminated by a PML

Jianxin Zhu¹, Zheqi Shen¹

1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China, Email: zjx@zju.edu.cn

Computer Modeling in Engineering & Sciences 2011, 71(4), 347-362. https://doi.org/10.3970/cmes.2011.071.347

Abstract

It is known that the perfectly matched layer (PML) is a powerful tool to truncate the unbounded domain. Recently, the PML technique has been introduced in the computation of nonlinear Schrödinger equations (NSE), in which the nonlinearity is separated by some efficient time-splitting methods. A major task in the study of PML is that the original equation is modified by a factor c which varies fast inside the layer. And a large number of grid points are needed to capture the profile of c in the discretization. In this paper, the possibility is discussed for using some nonuniform finite difference schemes in spatial discretization. It is proved that the uniform refinement inside the PML will cause spurious reflections at the interface, and therefore is invalid. As a remedy, a new method for the discretization is proposed by applying an additional coordinate transformation. This method effectively reduces the error caused by the discretization of PML, and it is quite useful in long time computation of time-dependent problems in open waveguides. These discussions are essential for the further research of the numerical solutions of NSE on unbounded domains.

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