In this paper, a family of arbitrarily high-order structure-preserving exponential Runge–Kutta methods are developed for the nonlinear Schrödinger 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.

在本文中，通过将标量辅助变量方法与指数Runge-Kutta方法相结合，为非线性Schrödinger方程开发了一系列任意保留高阶结构的指数Runge-Kutta方法。通过引入一个辅助变量，我们首先将原始模型转换成一个等效的系统，该系统同时接受质量和修正的节能规律。然后及时应用劳森法和辛格朗格-库塔法，我们推导出了一类质量和能量守恒的时间离散方案，它们在时间上是任意高阶的。数值实验旨在证明新提出的方案的准确性和有效性。