Optimized pairs of multidimensional ERKN methods with FSAL property for multi-frequency oscillatory systems

Abstract

For the integration of multi-frequency second-order oscillatory systems of differential equations, multidimensional extended Runge–Kutta–Nyström (MERKN) methods with FSAL property are investigated. Order conditions are presented. Phase properties are analysed and nonlinear stability of MERKN methods is proved. New practical MERKN methods with optimized phase and dissipation properties and embedded pairs of multidimensional MERKN methods are constructed. Numerical experiments are reported to show the high efficiency and robustness of our new methods compared with several well-known methods in the scientific literature.

Keywords:

• Multidimensional ERKN pair
• multi-frequency oscillatory system
• variable-stepsize
• phase-lag

2010 Mathematics Subject Classifications:

• 65L05
• 65L06

Acknowledgments

This work was partially supported by the Natural Science Foundation of China (NSFC) (No. 11571302, No. 11871268), the Natural Science Foundation of Shandong Province, China (No. ZR2018MA024), the Natural Science Foundation of Jiangsu Province, China (No. BK20171370), the Project of Shandong Province Higher Educational Science and Technology Program (No. J18KA247, No. J17KA190), the Foundation of Innovative Science and Technology for Youth in Universities of Shandong Province, China (No. 2019KJI001) and the project of Qingtan Scholars of Zaozhuang University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the Natural Science Foundation of China (NSFC) [No. 11571302, No. 11871268], the Natural Science Foundation of Shandong Province, China [No. ZR2018MA024], the Natural Science Foundation of Jiangsu Province, China [No. BK20171370], the Project of Shandong Province Higher Educational Science and Technology Program [No. J18KA247, No. J17KA190], the Foundation of Innovative Science and Technology for Youth in Universities of Shandong Province, China [No. 2019KJI001] and the project of Qingtan Scholars of Zaozhuang University.