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2009

Resonances in long time integration of semi linear Hamiltonian PDEs

Faou, Erwan, Grebert, Benoit

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We consider a class of Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part, and we analyze their numerical discretizations by symplectic methods when the initial value is small in Sobolev norms.

The goal of this work is twofold: First we show how standard approximation methods cannot in general avoid resonances issues, and we give numerical examples of pathological behavior for the midpoint rule and implicit-explicit integrators.
Such phenomena can be avoided by suitable truncations of the linear unbounded operator combined with classical splitting methods.
We then give a sharp bound for the cut-off depending on the time step.

Resonances in long time integration of semi linear Hamiltonian PDEs · Around

Built on

D. Bambusi and B. Grébert, Birkhoff normal form for PDE’s with tame modulus
2006
Earlier work this paper cites.
E. Hairer, C. Lubich and G. Wanner Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations
Original
2006
Earlier work this paper cites.
D. Bambusi, A birkhoff normal form theorem for some semilinear pdes , Hamiltonian Dynamical Systems and Applications, Springer, 2007, pp. 213–247
2007
Earlier work this paper cites.

Similar

G. Dujardin and E. Faou, Normal form and long time analysis of splitting schemes for the linear Schrödinger equation with small potential
2007
Cited alongside, same era.
B. Grébert, Birkhoff normal form and Hamiltonian PDEs
2007
Cited alongside, same era.
A. Debussche and E. Faou, Modified energy for split-step methods applied to the linear Schrödinger equation
Cited in the paper.
E. Faou, B. Grébert and E. Paturel, Birkhoff normal form and splitting methods for semi linear Hamiltonian PDEs. Part I: Finite dimensional discretization
Cited in the paper.
E. Faou, B. Grébert and E. Paturel, Birkhoff normal form and splitting methods for semi linear Hamiltonian PDEs. Part II: Abstract splitting
Cited in the paper.

Then

D. Cohen, E. Hairer and C. Lubich, Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations
2008
Later among the works it cites.
L. Gauckler and C. Lubich, Splitting integrators for nonlinear Schrödinger equations over long times
2009
Closest in time.

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