Multiple Relaxation Exponential Runge–Kutta Methods for the Nonlinear Schrödinger Equation

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2719-2744, December 2024.
Abstract. A novel family of high-order structure-preserving methods is proposed for the nonlinear Schrödinger equation. The methods are developed by applying the multiple relaxation idea to the exponential Runge–Kutta methods. It is shown that the multiple relaxation exponential Runge–Kutta methods can achieve high-order accuracy in time and preserve multiple original invariants at the discrete level. They are the first exponential-type methods that preserve multiple invariants. The number of invariants the methods preserve depends only on that of the relaxation parameters. Several numerical experiments are carried out to support the theoretical results and illustrate the effectiveness and efficiency of the proposed methods.