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

.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.Keywordsrelaxation techniqueexponential Runge–Kutta methodsstructure-preserving methodshigh-order accuracynonlinear Schrödinger equationMSC codes65M1265M2065M70