Unconditional optimal-order error estimates of linear relaxation compact difference scheme for the coupled nonlinear Schr\"odinger system

TL;DR

This paper introduces a high-order, mass- and energy-conserving numerical scheme for multi-dimensional coupled nonlinear Schrödinger systems, providing the first unconditional optimal error estimates in relevant norms.

Contribution

It develops a novel linear relaxation compact difference scheme with rigorous unconditional optimal error estimates for the multi-dimensional CNLS system, without coupling mesh restrictions.

Findings

Scheme conserves mass and energy exactly.

Achieves optimal-order error estimates unconditionally.

Demonstrates high accuracy and stability in long-term simulations.

Abstract

This paper presents a linear, decoupled, mass- and energy-conserving numerical scheme for the multi-dimensional coupled nonlinear Schr\"odinger (CNLS) system. The scheme combines the fourth-order compact difference approximation in space with the relaxation technique in a time-staggered mesh framework, solving the primal unknowns and introduced auxiliary relaxation variables sequentially with high efficiency and high-order accuracy. We show the unique solvability and discrete conservation laws of the developed scheme. In particular, for the first time, leveraging an auxiliary error equation system combined with the cut-off technique, optimal-order error estimates in the discrete H1-norm for the primal variables at the time nodes, and in the discrete L2-norm for the auxiliary relaxation variables at the intermediate time nodes, are rigorously proved without any coupling mesh conditions, which contribute to the primary theoretical contribution of this paper for multi-dimensional CNLS system. Numerical experiments demonstrate convincingly the strong performance of the proposed scheme in long-term simulations, maintaining both physical invariants and high-order accuracy.