Almost Symplectic Runge-Kutta Schemes for Hamiltonian Systems

Abstract

Symplectic Runge-Kutta schemes for integration of general Hamiltonian systems are implicit. In practice the implicit equations are often approximately solved based on the Contraction Mapping Principle, in which case the resulting integration scheme is no longer symplectic. In this note we prove that, under suitable conditions, the integration scheme based on an n-step successive approximation is $O(delta^{n+2})$ away from a symplectic scheme with $deltain(0,1)$. Therefore, this scheme is "almost" symplectic when n is large.