Section: New Results

Numerical analysis and simulation of heterogeneous systems

Participant : Xavier Antoine.

• In [10], we design some accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations in rectangular domains. We show the accuracy of the method thanks to simulations

• In [5], we design fast numerical and highly accurate methods for the computation of steady states and the dynamics of time or space-fractional Schrödinger equations.

• In [1], we design a numerical model of diffusion for the study of the properties of noble gases originating from volcanic eruptions.

• In [4], the deal with a multilevel Schwarz Waveform Relaxation (SWR) Domain Decomposition Method (DDM) for the Non Linear Schrödinger Equation (NLSE).

• In [6], we design a fast and pseudo spectral preconditioned conjugated gradient method for the computation of the steady states related to the Gross-Pitaevskii equation with non local dipolar interaction.

• In [3], we deal with fractional microlocal analysis for the obtention of asymptotic estimates for the convergence of Schwarz Waveform Relaxation (SWR) domain decomposition method; this study is done is the two dimensional quantum case.

• In [11], we design new methods of very high order for the computation of diffracted fields; these methods rely on a B-splines finite element method and are related to the isogeometric analysis.

• In [17], we deal with the numerical analysis of fast and accurate schemes for solving one-dimensional time-fractional nonlinear Schrödinger equations set with artificial boundaries.

• In [35], we obtain a close approximation of the optimal parameters for the convergence of domain decomposition methods for the Schrödinger equation.

• In [19], we compute an explicit approximation of the optimal parameters for the convergence of domain decomposition methods for the Schrödinger equation.

• In [21], we introduce an original method in order to integrate PML in a pseudospectral method for the computation of the dynamics of the Dirac equation. Some applications to lasers are given.

• In [20], we deal with the asymptotic analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation.