2009/12/15 by Erwan Faou, Faou, Erwan, Benoît Grébert +1 · 1 citation
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Dynamical Systems (math.DS) #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · doi:10.48550/arxiv.0912.2882
openalex publication_date 2009/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.