Sixth order non-dissipative C¹ spline collocation method for oscillatory ordinary initial value problems

Abstract

In this paper we construct a global method, based on quintic C ¹-spline, for the integration of first order ordinary initial value problems (IVPs) including stiff equations and those possessing oscillatory solutions as well. The method will be shown to be of order six and in particular is A-stable. Attention is also paid for the phase error (or dispersion) and it is proved that the method is dispersive and has dispersion order six with small phase-lag (compared with the extant methods having the same order (cf. [7])). Moreover, the method may be regarded as a continuous extension of the closed four-panel Newton–Cotes formula (NC4) (typically it is a continuous extension of an implicit Runge–Kutta method). In additiona priori error estimates, in the uniform norm, together with illustrative test examples will also be presented.

Keywords:

• Initial value problems
• Quintic spline
• Collocation Methods
• Absolute Stability
• A-stability
• Stiff equations
• Oscillating solutions
• Non-dissipative methods

AMS Classifications:

• 65L06
• 65D05

C.R. Categories:

• G.1.7