Error Estimate of Fourth-Order Compact Scheme for Linear Schrödinger Equations

Abstract

A fourth-order compact difference scheme is proposed for two-dimensional linear Schrödinger equations with periodic boundary conditions. By using the discrete energy method, it is proven that the difference scheme is uniquely solvable, unconditionally stable, and convergent. A maximum norm error estimate and thus an asymptotic expansion of the discrete solution are also obtained. Using the expansion of the difference solution, high-order approximations could be achieved by Richardson extrapolations. Extension to three-dimensional problems is also discussed. Numerical experiments are included to support the theoretical results, and comparisons with the Crank–Nicolson method are presented to show the effectiveness of our method.