Domain decomposition algorithms for the two dimensional nonlinear Schrödinger equation and simulation of Bose-Einstein condensates

International audienceIn this paper, we apply the optimized Schwarz method to the two dimensional nonlinear Schrödinger equation and extend this method to the simulation of Bose-Einstein condensates (Gross-Pitaevskii equation). We propose an extended version of the Schwartz method by introducing a preconditioned algorithm. The two algorithms are studied numerically. The experiments show that the preconditioned algorithm improves the convergence rate and reduces the computation time. In addition, the classical Robin condition and a newly constructed absorbing condition are used as transmission conditions

Absorbing boundary conditions for the two-dimensional Schrödinger equation with an exterior potential. Part II: Discretization and numerical results

International audienceFour families of ABCs where built in [5] for the two- dimensional linear Schr ̈odinger equation with time and space depen- dent potentials and for general smooth convex fictitious surfaces. The aim of this paper is to propose some suitable discretization schemes of these ABCs and to prove some semi-discrete stability results. Fur- thermore, the full numerical discretization of the corresponding initial boundary value problems is considered and simulations are provided to compare the accuracy of the different ABCs

Absorbing Boundary Conditions for General Nonlinear Schrödinger Equations

International audienceThis paper addresses the construction of different families of absorbing boundary conditions for the one- and two-dimensional Schrödinger equation with a general variable nonlinear potential. Various semi-discrete time schemes are built for the associated initial boundary value problems. Finally, some numerical simulations give a comparison of the various absorbing boundary conditions and associated schemes to analyze their accuracy and efficiency

High-order IMEX-spectral schemes for computing the dynamics of systems of nonlinear Schrödinger /Gross-Pitaevskii equations

International audienceThe aim of this paper is to build and validate some explicit high-order schemes, both in space and time, for simulating the dynamics of systems of nonlinear Schrödinger /Gross-Pitaevskii equations. The method is based on the combination of high-order IMplicit-EXplicit (IMEX) schemes in time and Fourier pseudo-spectral approximations in space. The resulting IMEXSP schemes are highly accurate, efficient and easy to implement. They are also robust when used in conjunction with an adaptive time stepping strategy and appear as an interesting alternative to time-splitting pseudo-spectral (TSSP) schemes. Finally, a complete numerical study is developed to investigate the properties of the IMEXSP schemes, in comparison with TSSP schemes, for one-and two-components systems of Gross-Pitaevskii equations

Absorbing Boundary Conditions for the Two-Dimensional Schrödinger Equation with an Exterior Potential. Part I: Construction and a priori Estimates

International audienceThe aim of this paper is to construct some classes of absorbing boundary conditions for the two-dimensional Schrödinger equation with a time and space varying exterior potential and for general convex smooth boundaries. The construction is based on asymptotics of the inhomogeneous pseudodifferential operators defining the related Dirichlet-to-Neumann operator. Furthermore, \textit{a priori} estimates are developed for the truncated problems with various increasing order boundary conditions. The effective numerical approximation will be treated in a second paper

Numerical Solution of Time-Dependent Nonlinear Schrödinger Equations Using Domain Truncation Techniques Coupled With Relaxation Scheme

International audienceThe aim of this paper is to compare different ways for truncating unbounded domains for solving general nonlinear one- and two-dimensional Schrödinger equations. We propose to analyze Complex Absorbing Potentials, Perfectly Matched Layers and Absorbing Boundary Conditions. The time discretization is made by using a semi-implicit relaxation scheme which avoids any fixed point procedure. The spatial discretization involves finite element methods. We propose some numerical experiments to compare the approaches