Extrapolation of symplectic methods for Hamiltonian problems

Abstract

We consider two modes of the extrapolation of symplectic and symmetric Runge–Kutta and related integrators over long time-intervals applied with constant stepsize. In the passive mode, we compute two solution sequences with stepsizes h and h/2 independently and perform extrapolation whenever output is required. In the active mode, we extrapolate at every step and propagate the extrapolated solution. We study and compare in detail both modes of extrapolation applied to the simple harmonic oscillator. We show that passive extrapolation will improve the accuracy of the numerical solution over the whole integration interval even though it destroys the linear error growth of the basic method and that active extrapolation exhibits linear error growth for the harmonic oscillator and, in general, yields higher accuracy than passive extrapolation. The error growth of integrations over long time-intervals is also studied in a more general setting. We obtain asymptotic error formulas for the periodic case and for integrable Hamiltonian systems where linear error growth has been established in the study of Calvo and Hairer [Appl. Numer. Math. 18 (1995) 95–105].