Backward Analysis Of Numerical Integrators And Symplectic Methods

Abstract

. A backward analysis of integration methods, whose numerical solution is a Pseries, is presented. Such methods include Runge-Kutta methods, partitioned Runge-Kutta methods and Nystrom methods. It is shown that the numerical solution can formally be interpreted as the exact solution of a perturbed differential system whose right-hand side is again a P-series. The main result of this article is that for symplectic integrators applied to Hamiltonian systems the perturbed differential equation is a Hamiltonian system too. The proofs use the one-to-one correspondence between rooted trees and the expressions appearing in the Taylor expansions of the exact and numerical solutions (elementary differentials). Key words. Backward analysis, Hamiltonian systems, Runge-Kutta methods, symplectic methods, P-series. 1. Introduction For the numerical solution of ordinary differential equations y 0 = f(y) (1:1) we consider one-step integrators such as Runge-Kutta methods or something similar. The ...