An explicit trigonometrically fitted Runge–Kutta method for stiff and oscillatory problems with two frequencies

ABSTRACT

In this paper, a new explicit fourth-order Runge–Kutta method adapted to oscillatory problems with two frequencies (μ and ν) is developed. The new method can integrate exactly the different harmonic oscillators $y^{\prime }=\mathrm{i}\mu y$ and $y^{\prime }=\mathrm{i}\nu y$ simultaneously. Numerical stability and phase properties of the new method are analysed. Numerical experiments are conducted to illustrate the high accuracy and competence of our new method in comparison with some highly effective methods from the recent literature.

KEYWORDS:

• Explicit Runge–Kutta method
• order condition
• stiff and oscillatory problem
• two-frequency trigonometrical fitting
• phase-lag

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65L04
• 65L05
• 65L12

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Yonglei Fang  http://orcid.org/0000-0002-5615-5971

Yanping Yang  http://orcid.org/0000-0002-8245-334X

Xiong You  http://orcid.org/0000-0002-0343-3878

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (NSFC) (grant Nos. 11571302, 11171155), and the Foundation of Scientific Research Project of Shandong Universities (grant No. J17KA190) and the Natural Science Foundation of Jiangsu Province (grant No. BK20171370) and Natural Science Foundation of Shandong Province (grant No. ZR201709190365).