Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrödinger-Poisson System (Communications in Computational Physics, 2013)
We study compact finite difference methods for the Schrödinger-Poissonequation in a bounded domain and establish their optimal error estimates under properregularity assumptions on wave function $ψ$ and external potential $V(x)$. The Crank-Nicolson compact finite difference method and the semi-implicit compact finite difference method are both of order $\mathcal{O}$($h^4$+$τ^2$) in discrete $l^2$, $H^1$ and $l^∞$ norms with meshsize $h$ and time step $τ$. For the errors of compact finite difference approximation tothe second derivative and Poisson potential are nonlocal, thus besides the standardenergy method and mathematical induction method, the key technique in analysis isto estimate the nonlocal approximation errors in discrete $l^∞$ and $H^1$ norm by discretemaximum principle of elliptic equation and properties of some related matrix. Alsosome useful inequalities are established in this paper. Finally, extensive numerical results are reported to support our error estimates of the numerical methods.