Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrödinger-Poisson System (Communications in Computational Physics, 2014)
We study the computation of ground states and time dependent solutionsof the Schrödinger-Poisson system (SPS) on a bounded domain in 2D (i.e. in two spacedimensions). On a disc-shaped domain, we derive exact artificial boundary conditionsfor the Poisson potential based on truncated Fourier series expansion in θ, and proposea second order finite difference scheme to solve the $r$-variable ODEs of the Fourier coefficients.The Poisson potential can be solved within $\mathcal{O}$($M NlogN$) arithmetic operationswhere $M,N$ 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-frogmethod are proposed to compute the ground state and dynamics respectively. Numericalresults are shown to confirm the accuracy and efficiency. Also we make it clearthat backward Euler sine pseudospectral (BESP) method in [33] can not be applied to2D SPS simulation.