Multiderivative Methods for Periodic Initial Value Problems

A family of two-step multiderivative methods is developed for the numerical solution of periodic initial value problems of the form ${\bf y}''(t) = {\bf f}(t,{\bf y})$ with initial conditions ${\bf y}(t_0 ) = {\bf y}_0 ,{\bf y}'(t_0 ) = {\bf y}'_0 $ given.The methods are developed by showing that the solution of the single test equation $y''(t) = - \lambda ^2 y(t)$, $\lambda $ real, satisfies a recurrence relation in which exponential functions are involved. The methods of the family are then derived by replacing the exponential terms by Pade approximants.The methods are analyzed for accuracy and periodicity properties; the error constants and intervals of periodicity are contained in an Appendix. Those Pade approximants for which the degree of the numerator do not exceed the degree of the denominator are seen to be P-stable Suitable predictor-corrector combinations of the methods are analyzed for use in PECE mode.The methods are compared with existing methods and they are tested on four problems from the...