574 research outputs found
Hamiltonian interpolation of splitting approximations for nonlinear PDEs
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
Quasi invariant modified Sobolev norms for semi linear reversible PDEs
We consider a general class of infinite dimensional reversible differential
systems. Assuming a non resonance condition on the linear frequencies, we
construct for such systems almost invariant pseudo norms that are closed to
Sobolev-like norms. This allows us to prove that if the Sobolev norm of index
of the initial data is sufficiently small (of order ) then
the Sobolev norm of the solution is bounded by during very long
time (of order with arbitrary). It turns out that this
theorem applies to a large class of reversible semi linear PDEs including the
non linear Schr\"odinger equation on the d-dimensional torus. We also apply our
method to a system of coupled NLS equations which is reversible but not
Hamiltonian.
We also notice that for the same class of reversible systems we can prove a
Birkhoff normal form theorem that in turn implies the same bounds on the
Sobolev norms. Nevertheless the technics that we use to prove the existence of
quasi invariant pseudo norms is much more simple and direct
Resonances in long time integration of semi linear Hamiltonian PDEs
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.
Using a new normal form result, we show the long time preservation of the
actions for such schemes for all values of the time step, provided the initial
continuous system does not exhibit resonant frequencies
Modified energy for split-step methods applied to the linear Schr\"odinger equation
We consider the linear Schr\"odinger equation and its discretization by
split-step methods where the part corresponding to the Laplace operator is
approximated by the midpoint rule. We show that the numerical solution
coincides with the exact solution of a modified partial differential equation
at each time step. This shows the existence of a modified energy preserved by
the numerical scheme. This energy is close to the exact energy if the numerical
solution is smooth. As a consequence, we give uniform regularity estimates for
the numerical solution over arbitrary long tim
Quasi-periodic solutions of the 2D Euler equation
We consider the two-dimensional Euler equation with periodic boundary
conditions. We construct time quasi-periodic solutions of this equation made of
localized travelling profiles with compact support propagating over a
stationary state depending on only one variable. The direction of propagation
is orthogonal to this variable, and the support is concentrated on flat strips
of the stationary state. The frequencies of the solution are given by the
locally constant velocities associated with the stationary state
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