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Title: Stability and error analysis for a second-order fast approximation of the one-dimensional schrödinger equation under absorbing boundary conditions
Authors: Li, B 
Zhang, J
Zheng, C
Issue Date: 2018
Source: SIAM journal on scientific computing, 2018, v. 40, no. 6, p. A4083-A4104
Abstract: A second-order Crank-Nicolson finite difference method, integrating a fast approximation of an exact discrete absorbing boundary condition, is proposed for solving the one-dimensional Schrödinger equation in the whole space. The fast approximation is based on Gaussian quadrature approximation of the convolution coefficients in the discrete absorbing boundary conditions. It approximates the time convolution in the discrete absorbing boundary conditions by a system of O(log 2 N) ordinary differential equations at each time step, where N denotes the total number of time steps. Stability and an error estimate are presented for the numerical solutions given by the proposed fast algorithm. Numerical experiments are provided, which agree with the theoretical results and show the performance of the proposed numerical method.
Keywords: Absorbing boundary condition
Error estimate
Fast algorithm
Gaussian quadrature
Schrödinger equation
Stability
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on scientific computing 
ISSN: 1064-8275
EISSN: 1095-7197
DOI: 10.1137/17M1162111
Rights: © 2018, Society for Industrial and Applied Mathematics.
Posted with permission of the publisher.
The following publication Li, B., Zhang, J., & Zheng, C. (2018). Stability and error analysis for a second-order fast approximation of the one-dimensional schrodinger equation under absorbing boundary conditions. SIAM Journal on Scientific Computing, 40(6), A4083-A4104 is available at https://dx.doi.org/10.1137/17M1162111
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