IPSO - 2015 - Annual activity report

IPSO

IPSO - 2015

Project-Team Ipso

Members

Overall Objectives

Research Program

Highlights of the Year

New Results

Uniformly accurate numerical schemes for highly-oscillatory Klein-Gordon and nonlinear Schrödinger equation
Higher-order averaging, formal series and numerical integration III: error bounds
Stroboscopic averaging for the nonlinear Schrödinger equation
Uniformly accurate time-splitting schemes for NLS in the semiclassical limit
Gyroaverage operator for a polar mesh
Asymptotic Preserving scheme for a kinetic model describing incompressible fluids
Numerical schemes for kinetic equations in the diffusion and anomalous diffusion limits. Part I: the case of heavy-tailed equilibrium
Comparison of numerical solvers for anisotropic diffusion equations arising in plasma physics
Hamiltonian splitting for the Vlasov-Maxwell equations
Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit
High-order Hamiltonian splitting for Vlasov-Poisson equations
Asymptotic Preserving numerical schemes for multiscale parabolic problems
Parallelization of an advection-diffusion problem arising in edge plasma physics using hybrid MPI/OpenMP programming
Numerical schemes for kinetic equations in the anomalous diffusion limit. Part II: degenerate collision frequency
Analysis of the Monte-Carlo error in a hybrid semi-Lagrangian scheme
Resonant time steps and instabilities in the numerical integration of Schrödinger equations
Collisions of almost parallel vortex filaments
On numerical Landau damping for splitting methods applied to the Vlasov-HMF model
A kinetic model for the transport of electrons in a graphen layer
Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime
The Interaction Picture method for solving the generalized nonlinear Schrödinger equation in optics
Nonlinear stability criteria for the HMF Model
Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement
Dimension reduction for anisotropic Bose-Einstein condensates in the strong interaction regime
Models of dark matter halos based on statistical mechanics: I. The classical King model
Numerical study of a quantum-diffusive spin model for two-dimensional electron gases
Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion
A regularity result for quasilinear stochastic partial differential equations of parabolic type
Diffusion limit for the radiative transfer equation perturbed by a Wiener process
Invariant measure of scalar first-order conservation laws with stochastic forcing
An integral inequality for the invariant measure of a stochastic reaction-diffusion equation
Estimate for $P_{t} D$ for the stochastic Burgers equation
Existence of the Fomin derivative of the invariant measure of a stochastic reaction-diffusion equation
Global behavior of $N$ competing species with strong diffusion: diffusion leads to exclusion
Randomized message-passing test-and-set

Partnerships and Cooperations

Dissemination

Bibliography

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Section: New Results

Analysis of the Monte-Carlo error in a hybrid semi-Lagrangian scheme

In [17] we consider Monte-Carlo discretizations of partial differential equations based on a combination of semi-lagrangian schemes and probabilistic representations of the solutions. The goal of this paper is twofold. First we give rigorous convergence estimates for our algorithm: In a simple setting, we show that under an anti-CFL condition on the time-step $δ t$ and on the mesh size $δ x$ and for a reasonably large number of independent realizations $N$ , we control the Monte-Carlo error by a term of order $𝒪 (\sqrt{δ t / N})$ . Then, we show various applications of the numerical method in very general situations (nonlinear, different boundary conditions, higher dimension) and numerical examples showing that the theoretical bound obtained in the simple case seems to persist in more complex situations.

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