Point-wise error estimates of two mass- and energy-preserving schemes for two-dimensional Schrödinger–Poisson equations

Abstract

This work presents two implicit and linear finite difference schemes that simultaneously preserve both mass and energy conservation properties for the two-dimensional Schrödinger–Poisson equations. The conservation, existence, uniqueness, as well as the convergence to the exact solution with the order $O({\tau }^2+h_x^2+h_y^2)$ in discrete $L^2$ and $L^{\infty }$ norms are established for these two schemes, where $\tau$ and $h_x,h_y$ represent temporal and spatial step sizes. In contrast to the existing analysis techniques that rely on an a priori $L^{\infty }$ estimate of numerical solutions or impose restrictions on initial data, our approaches guarantee the unconditional convergence for SP equations with both attractive and repulsive forces. Besides the standard energy method, our analytical framework employs the cut-off method for the implicit scheme and the mathematical induction argument for the linear scheme, where the “lifting” technique is utilized in the two schemes to eliminate the constraints on grid ratios. Numerical experiments are provided to illustrate discrete conservation properties and validate the achieved convergence results.