Commun. Comput. Phys., 16 (2014), pp. 764-780.

Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrodinger-Poisson System

Norbert J. Mauser 1, Yong Zhang 1*

1 Wolfgang Pauli Institute c/o Fak. Mathematik, University Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Received 11 August 2013; Accepted (in revised version) 14 March 2014
Available online 4 July 2014


We study the computation of ground states and time dependent solutions of the Schrodinger-Poisson system (SPS) on a bounded domain in 2D (i.e. in two space dimensions). On a disc-shaped domain, we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion in \theta, and propose a second order finite difference scheme to solve the r-variable ODEs of the Fourier coefficients. The Poisson potential can be solved within O(M N log N) arithmetic operations where M,N are the number of grid points in r-direction and the Fourier bases. Combined with the Poisson solver, a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively. Numerical results are shown to confirm the accuracy and efficiency. Also we make it clear that backward Euler sine pseudospectral (BESP) method in [33] can not be applied to 2D SPS simulation.

AMS subject classifications: 35Q55, 65M06, 65M22, 65T50, 81-08

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Key words: 2D Schrodinger-Poisson system, exact artificial boundary condition, backward Euler scheme, semi-implicit

*Corresponding author.
Email: (N. J. Mauser), (Y. Zhang)

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