High-order linearly implicit structure-preserving exponential integrators for the nonlinear Schr\"odinger equation

TL;DR

This paper introduces high-order, linearly implicit, energy-preserving exponential integrators for the nonlinear Schrödinger equation, combining scalar auxiliary variables, Fourier spectral methods, and Runge-Kutta schemes for efficient, accurate simulations.

Contribution

The paper develops a novel class of energy-preserving integrators that are high-order, linearly implicit, and specifically tailored for the nonlinear Schrödinger equation, improving efficiency and accuracy over existing methods.

Findings

The proposed schemes exactly conserve a modified energy.

Numerical results show superior performance compared to existing structure-preserving methods.

The methods require solving only linear equations with constant coefficients at each step.

Abstract

A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schr\"odinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation is first reformulated into an equivalent form which satisfies a quadratic energy. The spatial derivatives of the system are then approximated with the standard Fourier pseudo-spectral method. Subsequently, we apply the extrapolation technique/prediction-correction strategy to the nonlinear terms of the semi-discretized system and a linearized energy-conserving system is obtained. A fully discrete scheme is gained by further using the integrating factor Runge-Kutta method to the resulting system. We show that, under certain circumstances for the coefficients of a Runge-Kutta method, the proposed scheme can produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution and is extremely efficient in the sense that only linear equations with constant coefficients need to be solved at every time step. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other existing structure-preserving schemes.