Communications in Mathematical Sciences
- Commun. Math. Sci.
- Volume 1, Number 3 (2003), 501-556.
Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability
We propose a way to efficiently treat the well-known transparent boundary conditions for the Schrödinger equation. Our approach is based on two ideas: to write out a discrete transparent boundary condition (DTBC) using the Crank-Nicolson finite difference scheme for the governing equation, and to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for a rapid recursive calculation of the convolution.
We prove stability of the resulting initial-boundary value scheme, give error estimates for the considered approximation of the boundary condition, and illustrate the efficiency of the proposed method on several examples.
Commun. Math. Sci., Volume 1, Number 3 (2003), 501-556.
First available in Project Euclid: 21 August 2009
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 65M12: Stability and convergence of numerical methods 35Q40: PDEs in connection with quantum mechanics 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]
Arnold, Anton; Ehrhardt, Matthias; Sofronov, Ivan. Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability. Commun. Math. Sci. 1 (2003), no. 3, 501--556. https://projecteuclid.org/euclid.cms/1250880098