Nuria Reguera

Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

When a partial differential equation in an unbounded domain is solved numerically, it is necessary to introduce artificial boundary conditions. In this paper, a general class of absorbing boundary conditions is constructed for one-dimensional Schrodinger-type equations discretized in space by finite differences. For this, rational approximations to the transparent boundary conditions are used. We study the simplest case in detail, obtaining an estimate for the full discrete error and showing that the discrete problem is weakly unstable. Moreover, we show numerically that the discrete problems associated to higher order absorbing boundary conditions are more unstable. Several numerical experiments confirm the results previously obtained.

A high order finite element discretization with local absorbing boundary conditions of the linear Schrödinger equation

The goal of this paper is to obtain a high order full discretization of the initial value problem for the linear Schrodinger equation in a finite computational domain. For this we use a high order finite element discretization in space together with an adaptive implementation of local absorbing boundary conditions specifically obtained for linear finite elements, and a high order symplectic time integrator. The numerical results show that it is possible to obtain simultaneously a very good absorption at the boundary and a very small error in the interior of the computational domain.

Discrete absorbing boundary conditions for Schrödinger-type equations: practical implementation

Recently, some absorbing boundary conditions for Schrodingertype equations have been studied by Fevens, Jiang and Alonso-Mallo, and Reguera. These conditions make it possible to obtain a very high absorption at the boundary avoiding the nonlocality of transparent boundary conditions. However, the implementations used in the literature, where the boundary condition is chosen in a manual way in accordance with the solution or fixed independently of the solution, are not practical because of the small absorption. In this paper, a new practical adaptive implementation is developed that allows us to obtain automatically a very high absorption.

Simulation of coherent structures in nonlinear Schrödinger-type equations

This paper presents some numerical methods to simulate the evolution of coherent structures with small fluctuations, that appear as typical solutions of a class of nonintegrable nonlinear Schrodinger equations. The construction of the methods is particularly focused on two points: on one hand, the generation of the ground state profiles, to be used in the initial data of the simulations, combines a suitable spatial discretization with the resolution of a discrete variational problem. On the other hand, the approximation to leading parameters of these structures is controlled by the time integration. We compare different methods when simulating the evolution of initial ground state profiles and some initial data perturbed from them.

Avoiding order reduction when integrating nonlinear Schrödinger equation with Strang method

In this paper a technique is suggested to avoid order reduction when using Strang method to integrate nonlinear Schrodinger equation subject to time-dependent Dirichlet boundary conditions. The computational cost of this technique is negligible compared to that of the method itself, at least when the timestepsize is fixed. Moreover, a thorough error analysis is given as well as a modification of the technique which allows to conserve the symmetry of the method while retaining its second order.

A self-adjusting algorithm for solitary wave simulations

We introduce a practical algorithm to automate the simulations of solitary wave solutions of some nonlinear dispersive wave equations. The full discretization consists of a spatial discretization with a local basis and an invariant preserving time integrator. The algorithm includes a dynamic cleaning of dispersive tails and an automatic detection of the complete separation of the main pulses.

Long Time Numerical Approximation of Coherent-Structure Solutions of the Cubic Schrödinger Equation

The purpose of this work is to determine suitable numerical methods to simulate the evolution of coherent structures for the cubic nonlinear Schrodinger equation with Dirichlet boundary conditions on a finite one-dimensional interval. We consider different time integrators, some of them preserving one or two invariants of the problem. We show that the preservation of these invariants is essential for a good long time simulation.