Phase-fitted and amplification-fitted two-step hybrid methods for y″=f(x,y)y″=f(x,y)

This paper provides a theoretical framework for a new type of phase-fitted and amplification-fitted two-step hybrid (FTSH) methods which is introduced by the author in [H. Van de Vyver, A phase-fitted and amplification-fitted explicit two-step hybrid method for second-order periodic initial value problems, Internat. J. Modern Phys. C 17 (2006) 663–675]. The methods constitute a modification of dissipative two-step hybrid methods in the sense that two free parameters are added to eliminate the phase-lag and the amplification error. The methods are useful only when a good estimate of the frequency of the problem is known in advance. The parameters depend on the product of the estimated frequency and the stepsize. The algebraic order, zero-stability, stability and phase properties are examined. The theory is illustrated with sixth-order explicit FTSH methods. Numerical results carried out on an assortment of test problems show the relevance of the theory.