A new construction of modified equations for variational integrators

TL;DR

This paper introduces a novel method for constructing modified equations for variational integrators, specifically applied to the semilinear wave equation, by directly using the variational principle and modifying the symplectic structure.

Contribution

It presents a new construction approach for modified equations that maintains analytical properties and reduces regularity requirements in the context of variational integrators.

Findings

Successfully applied to the semilinear wave equation

Preserves analytical properties of the original system

Offers a systematic way to modify symplectic structure and Hamiltonian

Abstract

The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified equations with increasingly high frequencies which increase the regularity requirements on the analysis. In this paper, we consider the next order modified equations for the implicit midpoint rule applied to the semilinear wave equation to give a proof-of-concept of a new construction which works directly with the variational principle. We show that a carefully chosen change of coordinates yields a modified system which inherits its analytical properties from the original wave equation. Our method systematically exploits additional degrees of freedom by modifying the symplectic structure and the Hamiltonian together.