High-order mass- and energy-conserving methods for the coupled nonlinear Schrödinger equation

Abstract

A novel high-order numerical method, specifically designed to preserve the mass and energy invariants of the coupled nonlinear Schrödinger equation (CNLS) is introduced. This algorithm integrates Gauss collocation schemes for temporal discretization with finite element methods for spatial discretization, enhanced by a post-processing correction procedure that ensures mass and energy conservation at each time step. The theoretical framework for the proposed method rigorously establishes the existence, uniqueness, and high-order convergence of the numerical solutions. In particular, the error of our proposed scheme is proven to converge at the rate of $O({\tau }^{k+1}+h^p)$ in the $L^{\infty }(0,T;H^1)$-norm, where $\tau$ and $h$ denote the temporal and spatial step sizes, respectively, and $k$ and $p$ represent the degrees of the temporal and spatial finite element approximations, which can be arbitrarily high. The error analysis explicitly accounts for the cumulative effects of the post-processing correction across all time levels. A series of numerical experiments are presented to validate the proposed method, demonstrating its capacity to conserve mass and energy precisely, achieve high-order accuracy, and accurately capture soliton, bi-soliton and plane wave dynamics.