Backward error analysis for conjugate symplectic methods

TL;DR

This paper develops a systematic backward error analysis method for conjugate symplectic methods, revealing how these numerical schemes preserve modified geometric structures of Hamiltonian systems without relying on ansatz.

Contribution

It introduces a variational and symplectic technique-based backward error analysis that computes modified structures systematically for conjugate symplectic methods.

Findings

The method preserves modified symplectic forms and Hamiltonians.

It applies to symmetric linear multistep methods with matrix coefficients.

The approach does not depend on an ansatz, unlike previous methods.

Abstract

The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.