Analysis for parareal algorithms applied to Hamiltonian differential equations
M.J. Gander and E. Hairer
Abstract. Long-time integrations are an important issue in the numerical solution
of Hamiltonian systems. They are time consuming and it is natural
to consider the use of parallel architectures for reasons of efficiency.
In this context the parareal
algorithm has been proposed by several authors.
The present work is a theoretical study of the parareal algorithm when it is
applied to Hamiltonian differential equations. The idea of backward error
analysis is employed to get insight into the long-time behaviour of numerical approximations.
One of the main results is that convergence of
the parareal iterations restricts the length of the time window.
For nearly integrable systems its length is bounded by the square root
of the inverse of
the accuracy of the coarse integrator. The theoretical bounds are
confirmed by numerical experiments.
Key Words. Hamiltonian differential equations, parareal algorithm, parallel
architectures, symplectic methods, backward error analysis, long-time integration.