Christophe Besse

Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

Abstract In this paper, we begin with the nonlinear Schrodinger/Gross–Pitaevskii equation (NLSE/GPE) for modeling Bose–Einstein condensation (BEC) and nonlinear optics as well as other applications, and discuss their dynamical properties ranging from time reversible, time transverse invariant, mass and energy conservation, and dispersion relation to soliton solutions. Then, we review and compare different numerical methods for solving the NLSE/GPE including finite difference time domain methods and time-splitting spectral method, and discuss different absorbing boundary conditions. In addition, these numerical methods are extended to the NLSE/GPE with damping terms and/or an angular momentum rotation term as well as coupled NLSEs/GPEs. Finally, applications to simulate a quantized vortex lattice dynamics in a rotating BEC are reported.

Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional Schrödinger equation

This paper addresses the problem of the construction of stable approximation schemes for the one-dimensional linear Schrodinger equation set in an unbounded domain. After a study of the initial boundary-value problem in a bounded domain with a transparent boundary condition, some unconditionally stable discretization schemes are developed for this kind of problem. The main difficulty is linked to the involvement of a fractional integral operator defining the transparent operator. The proposed semi-discretization of this operator yields with a very different point of view the one proposed by Yevick, Friese and Schmidt [J. Comput. Phys. 168 (2001) 433]. Two possible choices of transparent boundary conditions based on the Dirichlet-Neumann (DN) and Neumann-Dirichlet (ND) operators are presented. To preserve the stability of the fully discrete scheme, conform Galerkin finite element methods are employed for the spatial discretization. Finally, some numerical tests are performed to study the respective accuracy of the different schemes.

Order Estimates in Time of Splitting Methods for the Nonlinear Schrödinger Equation

In this paper, we consider the nonlinear Schrodinger equation

Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

u_t+i\Delta u -F(u)=0

A Relaxation Scheme for the Nonlinear Schrödinger Equation

in two dimensions. We show, by an operator-theoretic proof, that the well-known Lie and Strang formulae (which are splitting methods) are approximations of the exact solution of order 1 and 2 in time.

Artificial boundary conditions for one-dimensional cubic nonlinear Schrödinger equations

This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schrödinger equation in a bounded domain Ω with artificial boundary conditions set on the arbitrarily shaped boundary of Ω. These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. After having proved the well-posedness of the continuous truncated initial boundary value problem, a semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and its stability is provided. Next, the full discretization is realized by way of a standard finite-element method to preserve the stability of the scheme. Some numerical simulations are given to illustrate the effectiveness and flexibility of the method.

CONSTRUCTION, STRUCTURE AND ASYMPTOTIC APPROXIMATIONS OF A MICRODIFFERENTIAL TRANSPARENT BOUNDARY CONDITION FOR THE LINEAR SCHRÖDINGER EQUATION

In this paper, we present a new numerical scheme for the nonlinear Schrodinger equation. This is a relaxation-type scheme that avoids solving for nonlinear systems and preserves density and energy. We give convergence results for the semidiscretized version of the scheme and perform several numerical experiments.

Absorbing boundary conditions for the one-dimensional Schrödinger equation with an exterior repulsive potential

This paper addresses the construction of nonlinear integro-differential artificial boundary conditions for one-dimensional nonlinear cubic Schrodinger equations. Several ways of designing such conditions are provided and a theoretical classification of their accuracy is given. Semidiscrete time schemes based on the method developed by Duran and Sanz-Serna [IMA J. Numer. Anal. 20 (2000), pp. 235-261] are derived for these unusual boundary conditions. Stability results are stated and several numerical tests are performed to analyze the capacity of the proposed approach.

NUMERICAL STUDY OF ELLIPTIC-HYPERBOLIC DAVEY–STEWARTSON SYSTEM: DROMIONS SIMULATION AND BLOW-UP

A transparent boundary condition for the two-dimensional linear Schrodinger equation is constructed through a microlocal approximation of the operator associating the Dirichlet data to the Neumann one in a M-quasi hyperbolic region. Several quasi-analytic characterization results concerning the asymptotic expansion of the total symbol of this operator in a subclass of inhomogeneous symbols with a quasi-polynomial-like structure are stated. In particular, a high-frequency control giving the behavior of these symbols is precised. It highlights the way of how to derive some consistent asymptotic artificial boundary conditions involving fractional derivatives with respect to the time variable by approximating the micro-transparent condition in the high-frequency regime. These approximate conditions are local according to the space variable and should lead to some efficient and accurate numerical simulations if they are used to truncate the unbounded domain of propagation.

Absorbing Boundary Conditions for General Nonlinear Schrödinger Equations

Mathematical constructions and comparisons of accurate absorbing boundary conditions for the one-dimensional Schrodinger equation with a general variable repulsive potential are developed. Stable semi-discretization schemes are built for the associated initial boundary value problems. Finally, some numerical simulations give a comparison of the various absorbing boundary conditions and show that they yield accurate computations.