Energy-Conserving Explicit Relaxed Runge–Kutta Methods for the Fractional Nonlinear Schrödinger Equation Based on Scalar Auxiliary Variable Approach

Abstract

In this paper, we present a novel explicit structure-preserving numerical method for solving nonlinear space-fractional Schrödinger equations based on the concept of the scalar auxiliary variable approach. Firstly, we convert the equations into an equivalent system through the introduction of a scalar variable. Subsequently, a semi-discrete energy-preserving scheme is developed by employing a fourth-order fractional difference operator to discretize the equivalent system in spatial direction, and obtain the fully discrete version by using an explicit relaxed Runge–Kutta method for temporal integration. The proposed method preserves the energy conservation property of the space-fractional nonlinear Schrödinger equation and achieves high accuracy. Numerical experiments are carried out to verify the structure-preserving qualities of the proposed method.