Hamiltonian interpolation of splitting approximations for nonlinear PDEs

TL;DR

This paper develops a backward error analysis for splitting methods applied to semi-linear Hamiltonian PDEs, showing that the numerical solution closely follows a modified Hamiltonian flow over long times under certain conditions.

Contribution

It introduces a modified Hamiltonian framework for splitting methods on nonlinear PDEs, establishing long-time accuracy and stability results under non-resonance conditions.

Findings

Numerical flow remains close to the exact flow over exponentially long times.

High-order modified splitting schemes satisfy the non-resonance condition.

Standard splitting schemes are accurate over long times under CFL conditions.

Abstract

We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni- formly integrable with respect to an eigenbasis of the linear operator (typically the Fourier basis). We show the existence of a modified interpolated Hamiltonian equation whose exact solution coincides with the discrete flow at each time step over a long time depending on a non resonance condition satisfied by the stepsize. We introduce a class of modified splitting schemes fulfilling this condition at a high order and prove for them that the numerical flow and the continuous flow remain close over exponentially long time with respect to the step size. For stan- dard splitting or implicit-explicit scheme, such a backward error analysis result holds true on a time depending on a cut-off condition in the high frequencies (CFL condition). This analysis is valid in the case where the linear operator has a discrete (bounded domain) or continuous (the whole space) spectrum.