On an asymptotic method for computing the modified energy for symplectic methods

TL;DR

This paper enhances an existing algorithm for calculating the modified energy in symplectic methods, achieving arbitrarily high accuracy via Richardson extrapolation and confirming exponential error decay through theoretical analysis and numerical tests.

Contribution

It reformulates Skeel et al.'s algorithm as a Richardson extrapolation scheme, enabling high-order accuracy and providing error estimates for the shadow energy computation.

Findings

High-order accuracy achieved through Richardson extrapolation.

Error estimates confirm exponential smallness of energy drift.

Numerical examples validate theoretical results.

Abstract

We revisit an algorithm by Skeel et al. for computing the modified, or shadow, energy associated with the symplectic discretization of Hamiltonian systems. By rephrasing the algorithm as a Richardson extrapolation scheme arbitrary high order of accuracy is obtained, and provided error estimates show that it does capture the theoretical exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory.