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.

李东方，华中科技大学数学与统计学院教授，博导，华中卓越学者，国家级高层次青年人才。主持国家级课题7项。主要从事微分方程数值解、机器学习和信号处理等领域的研究工作。尤其在微分方程保结构算法和分数阶微分方程的高效数值算法和理论上取得一些有意义的进展。