A decoupled linear, mass- and energy-conserving relaxation-type high-order compact finite difference scheme for the nonlinear Schrödinger equation

Abstract

In this paper, a relaxation-type high-order compact finite difference (RCFD) scheme is proposed for the one-dimensional nonlinear Schrödinger equation. More specifically, the relaxation approach combined with the Crank-Nicolson formula is utilized for time discretization, and fourth-order compact difference method is applied for space discretization. The scheme is linear, decoupled, and can be solved sequentially with respect to the primal and relaxation variables, which avoids solving large-scale nonlinear algebraic systems resulting in fully implicit numerical schemes. Furthermore, the developed scheme is shown to preserve both mass and energy at the discrete level. Most importantly, with the help of a discrete elliptic projection and a cut-off numerical technique, the existence and uniqueness of the high-order RCFD scheme are ensured, and unconditional optimal-order error estimate in discrete maximum-norm is rigorously established. Finally, several numerical experiments are given to support the theoretical findings, and comparisons with other methods are also presented to show the efficiency and effectiveness of our method.