Hamiltonian Interpolation of Splitting Approximations for Nonlinear PDEs

We consider a wide class of semi linear Hamiltonian partial differential equations and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical trajectory remains at least uniformly 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 standard 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 analy...

• Type: Journal
• Year: 2011
• Where: FOCM
• Authors: Erwan Faou, Benoît Grebert
• Keywords: Explicit Scheme, FOCM 2011, Formal Methods, Fourier Basis, Hamiltonian Equation