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Title: An efficient second-order finite difference method for the one-dimensional schrodinger equation with absorbing boundary conditions
Authors: Li, BY 
Zhang, JW
Zheng, CX
Keywords: Schrodinger equation
Absorbing boundary condition
Convolution quadrature
Pade approximation
Fast algorithm
Error estimate
Issue Date: 2018
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on numerical analysis, 2018, v. 56, no. 2, p. 766-791 How to cite?
Journal: SIAM journal on numerical analysis 
Abstract: A stable and convergent second-order fully discrete finite difference scheme with efficient approximation of the exact absorbing boundary conditions is proposed to solve the Cauchy problem of the one-dimensional Schrodinger equation. Our approximation is based on the Pade expansion of the square root function in the complex plane. By introducing a constant damping term to the governing equation and modifying the standard Crank-Nicolson implicit scheme, we show that the fully discrete numerical scheme is unconditionally stable if the order of Pade expansion is chosen from our criterion. In this case, an optimal-order asymptotic error estimate is proved for the numerical solutions. Numerical examples are provided to support the theoretical analysis and illustrate the performance of the proposed numerical scheme.
ISSN: 0036-1429
EISSN: 1095-7170
DOI: 10.1137/17M1122347
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