2,615 research outputs found
Numerical algorithms for Schrödinger equation with artificial boundary conditions
We consider a one-dimensional linear Schrödinger problem defined on an infinite domain and approximated by the Crank-Nicolson type finite difference scheme. To solve this problem numerically we restrict the computational domain by introducing the reflective, absorbing or transparent artificial boundary conditions. We investigate the conservativity of the discrete scheme with respect to the mass and energy of the solution. Results of computational experiments are presented and the efficiency of different artificial boundary conditions is discussed
Compact high order finite difference schemes for linear Schrödinger problems on non-uniform meshes
In the present paper a general technique is developed for construction
of compact high-order finite difference schemes to approximate Schrödinger
problems on nonuniform meshes. Conservation of the finite difference schemes
is investigated. Discrete transparent boundary conditions are constructed for
the given high-order finite difference scheme. The same technique is applied
to construct compact high-order approximations of the Robin and Szeftel type
boundary conditions. Results of computational experiments are presente
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Compact high order finite difference schemes for linear Schrödinger problems on non-uniform meshes
In the present paper a general technique is developed for construction
of compact high-order finite difference schemes to approximate Schrödinger
problems on nonuniform meshes. Conservation of the finite difference schemes
is investigated. Discrete transparent boundary conditions are constructed for
the given high-order finite difference scheme. The same technique is applied
to construct compact high-order approximations of the Robin and Szeftel type
boundary conditions. Results of computational experiments are presente
Comparison of the continuous, semi-discrete and fully-discrete transparent boundary conditions (TBC) for the parabolic wave equation 1. Theory
For the simulation of the propagation of optical waves in open wave guiding structures of integrated optics the parabolic approximation of the scalar wave equation is commonly used. This approach is commonly termed the beam propagation method (BPM). It is of paramount importance to have well-performing transparent boundary conditions applied on the boundaries of the finite computational window, to enable the superfluous portion of the propagating wave to radiate away from the wave guiding structure. Three different formulations (continuous, semi-discrete and fully-discrete) of the non-local transparent boundary conditions are described and compared here
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Implementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh-Weyl theory
A new approach for simulating the solution of the time-dependent
Schrödinger equation with a general variable potential will be proposed. The
key idea is to approximate the Titchmarsh-Weyl m-function (exact
Dirichlet-to-Neumann operator) by a rational function with respect to a
suitable spectral parameter. With the proposed method we can overcome the
usual high-frequency restriction for absorbing boundary conditions of general
variable potential problems. We end up with a fast computational algorithm
for absorbing boundary conditions that are accurate for the full frequency
ban
Implementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh--Weyl theory
A new approach for simulating the solution of the time-dependent Schrödinger equation with a general variable potential will be proposed. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to a suitable spectral parameter. With the proposed method we can overcome the usual high-frequency restriction for absorbing boundary conditions of general variable potential problems. We end up with a fast computational algorithm for absorbing boundary conditions that are accurate for the full frequency band
Discrete transparent boundary conditions for the mixed KDV-BBM equation
International audienceIn this paper, we consider artificial boundary conditions for the linearized mixed Korteweg-de Vries (KDV) Benjamin-Bona-Mahoney (BBM) equation which models water waves in the small amplitude, large wavelength regime. Continuous (respectively discrete) artificial boundary conditions involve non local operators in time which in turn requires to compute time convolutions and invert the Laplace transform of an analytic function (respectively the Z-transform of an holomor-phic function). In this paper, we propose a new, stable and fairly general strategy to carry out this crucial step in the design of transparent boundary conditions. For large time simulations, we also introduce a methodology based on the asymptotic expansion of coefficients involved in exact direct transparent boundary conditions. We illustrate the accuracy of our methods for Gaussian and wave packets initial data
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