Explicit high-order conservative exponential time differencing Runge-Kutta schemes for the two-dimensional nonlinear Schrödinger equation

Abstract

In this paper, we develop a class of explicit energy-preserving Runge-Kutta schemes for solving the nonlinear Schrödinger equation based on the projection technique and the exponential time differencing method. First, we reformulate the equation to an equivalent system that possesses new quadratic energy via introducing an auxiliary variable. Then, we construct a family of fully discrete exponential time differencing schemes which have better stability by using the Runge-Kutta method and the Fourier-pseudo spectral method to approximate the system in time and space, respectively. Subsequently, energy-preserving schemes are derived by combining the proposed explicit schemes and the projection technique, and the stability result is given. Finally, extensive numerical examples are presented to confirm the constructed schemes have high accuracy, energy-preserving and effectiveness in long time simulation.