2024年6月28日，Elsevier 旗下top期刊《Journal of Computational Physics》（影响因子：3.8） 在线发表了云南师范大学数学学院梅丽杰教授团队的最新研究成果《Embedded exponential Runge–Kutta–Nyström methods for highly oscillatory Hamiltonian systems》。云南师范大学数学学院为第一单位，通讯作者为云南师范大学数学学院梅丽杰教授。合作单位为西安交通大学数学与统计学院

（School of Mathematics and Statistics, Xi'an Jiaotong  University）蒋耀林教授。

https://www.sciencedirect.com/science/article/abs/pii/S0021999124004704

Abstract

Exponential/extended Runge–Kutta–Nyström (ERKN) methods have been validated by numerous theoretical and numerical results to be more efficient and suitable than classical Runge–Kutta–Nyström (RKN) methods in dealing with highly oscillatory Hamiltonian systems. Once numerical solutions are required to achieve a prescribed local error tolerance for practical problems, the embedded ERKN pairs with adaptive stepsize are needed. Given the higher efficiency of high-order methods than low-order methods, this paper focuses on high-order embedded ERKN pairs. Using the particular mapping from RKN methods into ERKN methods, we establish an approach to constructing high-order embedded ERKN pairs. By diagonalizing the frequency matrix, an implementation algorithm with less calculation amount than the direct calculation procedure is proposed for high-dimensional problems. In addition, the dispersion and dissipation of the ERKN pairs ERKN4(3) and ERKN8(6) are analyzed. Furthermore, the application of ERKN pairs to an important highly oscillatory problem, i.e., the wave equation prescribed with different boundary conditions is discussed. For the periodic boundary condition, a special implementation algorithm incorporated with FFT techniques is presented, which decreases the calculation amount in one time step from �(�2) to �(�log⁡�). Finally, numerical results with the FPU problem, the Klein–Gordon equation in the nonrelativistic limit regime, and a two-dimensional wave equation demonstrate the superiority of ERKN pairs over classical RKN pairs.

扩展阅读：

梅丽杰教授加盟云南师范大学数学学院

https://math.ynnu.edu.cn/info/1027/2027.htm

姓   名：梅丽杰，云南师范大学数学学院教授，江西省科技创新杰出青年人才培养计划，

电子邮箱：bxhanm@126.com

研究方向及招生情况

l 研究领域：微分方程数值解   l 招生专业：计算数学（硕士）

工作经历

2022.12——至今，云南师范大学，教授

2016.06——2022.12，上饶师范学院，讲师、副教授、教授

教育经历

2013.09——2016.06，南京大学，数学系，数学，博士研究生

2010.09——2013.06，南昌大学，理学院，计算数学，硕士研究生

2006.09——2010.07，南昌大学，理学院，信息与计算科学，本科

学术兼职

中国数学会会员、中国天文学会会员、江西省天文学会理事、美国数学评论评论员

科研项目

2022.01—2025.12，后牛顿N体问题的长期数值动力学相关问题研究，国家自然科学基金地区项目（12163003），37万，主持

2019.01—2021.12，高振荡保守/耗散系统的指数型保结构算法，国家自然科学基青年科学基金项目（11801377），23万，主持

2019.07—2022.06，江西省科技创新杰出青年人才培养计划（现江西省主要学科学术和技术带头人培养计划青年人才项目，20192BCBL23030），江西省科技厅，30万，主持

2019.07—2022.06，后牛顿拉格朗日系统的保结构算法理论及应用，江西省自然科学基金青年基金重点项目（现杰出青年基金项目，20192ACBL21053），20万，主持

2017.07—2019.06，高振荡哈密尔顿波方程的保结构算法，江西省自然科学基金青年基金项（20171BAB211005），6万，主持

代表性论文

1. Lijie Mei; Li Huang; Xinyuan Wu*; Energy-preserving continuous-stage exponential Runge--Kutta integrators for efficiently solving Hamiltonian systems, SIAM Journal on Scientific Computing, 2022, 44(3): A1092--A1115.

2. Lijie Mei; Li Huang; Xinyuan Wu*; A unified framework for the study of high-order energy-preserving integrators for solving Poisson systems, Journal of Computational Physics, 2022, 450: 110822.

3. Lijie Mei; Li Huang; Xinyuan Wu*; Energy-preserving exponential integrators of arbitrarily high order for conservative or dissipative systems with highly oscillatory solutions, Journal of Computational Physics, 2021, 442: 110429.

4. Li Huang; Lijie Mei*; Energy-preserving integrators for post-Newtonian Lagrangian dynamics, Astrophysical Journal Supplement Series, 2020, 251(1): 8.

5. Li Huang; Lijie Mei*; Symplectic integrators for post-Newtonian Lagrangian dynamics, Physical Review D, 2019, 100(2): 024057.

6. Lijie Mei*; Li Huang; Shixiang Huang; Exponential integrators with quadratic energy preservation for linear Poisson systems, Journal of Computational Physics, 2019, 387: 446--454.

7. Lijie Mei*; Li Huang; Reliability of Lyapunov characteristic exponents computed by the two-particle method, Computer Physics Communications, 2018, 224: 108--118.

8. Lijie Mei; Changying Liu; Xinyuan Wu*; An essential extension of the finite-energy condition for extended Runge--Kutta--Nyström integrators when applied to nonlinear wave equations, Communications in Computational Physics, 2017, 22(3): 742--764.

9. Lijie Mei; Xinyuan Wu*; Symplectic exponential Runge--Kutta methods for solving nonlinear Hamiltonian systems, Journal of Computational Physics, 2017, 338: 67--584.

10. Lijie Mei; Xinyuan Wu*; The construction of arbitrary order ERKN methods based on group theory for solving oscillatory Hamiltonian systems with applications, Journal of Computational Physics, 2016, 323: 171--190.