Trigonometrically fitted multi-step hybrid methods for oscillatory special second-order initial value problems

ABSTRACT

In this paper, trigonometrically fitted multi-step hybrid (TFMSH) methods for the numerical integration of oscillatory special second-order initial value problems are proposed and studied. TFMSH methods inherit the frame of multi-step hybrid (MSH) methods and integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set $\{\exp (\mathrm{i}wt),\exp (-\mathrm{i}wt\left)\right\}$ or equivalently the set $\{\cos (wt),\sin (wt\left)\right\}$, where w represents an approximation of the main frequency of the problem. The corresponding order conditions are given and two explicit TFMSH methods with order six and seven, respectively, are constructed. Stability of the new methods is examined and the corresponding regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature.

Keywords:

• Trigonometrically fitted methods
• multi-step hybrid methods
• order conditions
• explicit methods
• oscillatory special second-order initial value problems

2010 AMS Subject Classifications:

• 65L05
• 65L06

Acknowledgements

The authors are grateful to the two anonymous reviewers for their valuable suggestions, which help improve this paper significantly.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported in part by the Natural Science Foundation of China under Grant Nos: [11401164, 11201113, and 11401163] and by Hebei Natural Science Foundation of China under Grant Nos: [A2014205136, A2014205133, and A2014205149].