[Paper Review] Hamiltonian interpolation of splitting approximations for nonlinear PDEs

Erwan Faou, Benoît Grébert|arXiv (Cornell University)|Dec 15, 2009

Numerical methods for differential equations3 references4 citations

TL;DR

This paper establishes Hamiltonian interpolation for splitting integrators applied to nonlinear Hamiltonian PDEs with polynomial nonlinearities, showing that the numerical flow of a modified Hamiltonian closely matches the exact flow over exponentially long times. By constructing a modified Hamiltonian via backward error analysis that accounts for spectral filtering, the method ensures long-time preservation of energy and regularity under non-resonance and integrability 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.

Motivation & Objective

• To extend backward error analysis to semi-linear Hamiltonian PDEs with unbounded linear operators, overcoming the blow-up of constants in finite-dimensional estimates.
• To establish the existence of a modified Hamiltonian whose exact flow matches the discrete splitting scheme at each time step over exponentially long times.
• To analyze splitting schemes (e.g., Lie-Trotter) for nonlinear Schrödinger and wave equations under spectral filtering to ensure long-time stability.
• To prove that numerical solutions remain close to the continuous flow under non-resonance and uniform integrability conditions in Fourier space.
• To derive quantitative bounds on energy preservation and regularity over exponentially long times depending on step size.

Proposed method

• Formal backward error analysis constructs a modified Hamiltonian $ H_h $ as a power series in $ h $, using iterated Poisson brackets between the linear and nonlinear parts of the Hamiltonian.
• A spectral filtering function $ \alpha_h(x) $ is introduced to regularize the linear operator, ensuring boundedness of the modified Hamiltonian and avoiding unbounded eigenvalues.
• The method applies to both discrete (torus) and continuous (whole space) spectra, using Fourier-based representations and pseudo-spectral methods.
• Key estimates rely on zero-momentum conditions and $ L^1 $-boundedness of the solution in Fourier space to control growth in high-frequency modes.
• The recursive solution of the modified Hamiltonian equation uses a Lie-Poisson bracket formalism with multi-indexed integrals over frequency configurations.
• Non-resonance conditions $ |\Lambda(\boldsymbol{\tau})| \geq 2\pi - \delta $ are imposed to ensure convergence and exponential smallness of error terms.

Experimental results

Research questions

• RQ1 Can a modified Hamiltonian be constructed such that the exact flow of $ H_h $ coincides with the discrete splitting map at each time step?
• RQ2 Under what conditions does the numerical solution of a splitting scheme remain close to the continuous flow over exponentially long times?
• RQ3 How can spectral filtering be used to stabilize splitting methods for PDEs with unbounded linear operators?
• RQ4 What role does uniform integrability in Fourier space play in ensuring long-time accuracy?
• RQ5 To what extent can energy and regularity be preserved over exponentially long times using this modified Hamiltonian framework?

Key findings

• The numerical flow of the splitting scheme coincides with the exact flow of a modified Hamiltonian $ H_h $ up to exponentially small error $ \mathcal{O}(\exp(-c h^{-1/2})) $ for $ nh \leq \exp(c h^{-1/2}) $.
• For the cubic NLS with $ \alpha_h(x) = \sqrt{h} \arctan(x/\sqrt{h}) $, the modified Hamiltonian $ H_h^{(1)} $ preserves the initial energy up to $ \mathcal{O}(h) $ over exponentially long times.
• The solution remains bounded in $ H^1 $-norm over exponentially long times if the initial data is in $ H^1(\mathbb{R}^3) $, with control on high-frequency components via the filter.
• If the solution remains in $ B^s_M $ with $ s \geq 3 $, the original Hamiltonian $ H $ is preserved up to $ \mathcal{O}(h) $ over exponentially long times.
• The non-resonance condition $ |\Lambda(\boldsymbol{\tau})| \leq 2\pi - \delta $ ensures convergence of the formal series and exponential stability of the modified system.
• The method applies uniformly to both discrete and continuous spectra, including the whole space $ \mathbb{R}^d $, under appropriate spectral filtering.