Mass- and energy-preserving exponential Runge-Kutta methods for the nonlinear Schr\"odinger equation

TL;DR

This paper develops high-order exponential Runge-Kutta methods that preserve mass and energy for the nonlinear Schrödinger equation by using auxiliary variables and structure-preserving techniques.

Contribution

It introduces a novel combination of scalar auxiliary variable approach with exponential Runge-Kutta methods to achieve high-order, structure-preserving schemes for the nonlinear Schrödinger equation.

Findings

Methods are arbitrarily high-order in time.

Numerical experiments confirm accuracy and effectiveness.

Schemes preserve mass and modified energy.

Abstract

In this paper, a family of arbitrarily high-order structure-preserving exponential Runge-Kutta methods are developed for the nonlinear Schr\"odinger equation by combining the scalar auxiliary variable approach with the exponential Runge-Kutta method. By introducing an auxiliary variable, we first transform the original model into an equivalent system which admits both mass and modified energy conservation laws. Then applying the Lawson method and the symplectic Runge-Kutta method in time, we derive a class of mass- and energy-preserving time-discrete schemes which are arbitrarily high-order in time. Numerical experiments are addressed to demonstrate the accuracy and effectiveness of the newly proposed schemes.