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Erwan Faou

  1. Weak backward error analysis for SDEs.

    Ср, 05/04/2011 - 15:01 — SomNambul
    Authors: Erwan Faou, Arnaud Debussche
    Subjects: Numerical Analysis
    Abstract

    We consider numerical approximations of stochastic differential equations by
    the Euler method. In the case where the SDE is elliptic or hypoelliptic, we
    show a weak backward error analysis result in the sense that the generator
    associated with the numerical solution coincides with the solution of a
    modified Kolmogorov equation up to high order terms with respect to the
    stepsize.

  2. A Nekhoroshev type theorem for the nonlinear Schr\"odinger equation on the d-dimensional torus..

    Пт, 03/26/2010 - 16:01 — SomNambul
    Authors: Erwan Faou, Benoit Grebert
    Subjects: Dynamical Systems
    Abstract

    We prove a Nekhoroshev type theorem for the nonlinear Schr\"odinger equation
    $$ iu_t=-\Delta u+V\star u+\partial_{\bar u}g(u,\bar u)\, \quad x\in \T^d, $$
    where $V$ is a typical smooth potential and $g$ is analytic in both variables.
    More precisely we prove that if the initial datum is analytic in a strip of
    width $\rho>0$ with a bound on this strip equals to $\eps$ then, if $\eps$ is
    small enough, the solution of the nonlinear Schr\"odinger equation above
    remains analytic in a strip of width $\rho/2$ and bounded on this strip by
    $C\eps$ during very long time of order $ \eps^{-\alpha|\ln

  3. Hamiltonian interpolation of splitting approximations for nonlinear PDEs.

    Ср, 12/16/2009 - 16:01 — SomNambul
    Authors: Erwan Faou, Benoit Grebert
    Subjects: Numerical Analysis
    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.

  4. Quasi invariant modi?ed Sobolev norms for semi linear reversible PDEs.

    Пнд, 08/24/2009 - 13:10 — SomNambul
    Authors: Erwan Faou, Benoit Grebert
    Subjects: Mathematical Physics
    Abstract

    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
    $s$ of the initial data $z_0$ is sufficiently small (of order $\epsilon$) then
    the Sobolev norm of the solution is bounded by $2\epsilon$ during very long
    time (of order $\epsilon^{-r}$ with $r$ arbitrary).

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