476 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
ABC Method and Fractional Momentum Layer for the FDTD Method to Solve the Schrödinger Equation on Unbounded Domains
The finitedifference timedomain (FDTD) method and its generalized variant (GFDTD) are efficient numerical tools for solving the linear and nonlinear Schrödinger equations because not only are they explicit, allowing parallelization, but they also provide highorder accuracy with relatively inexpensive computational costs. In addition, the GFDTD method has a relaxed stability condition when compared to the original FDTD method. It is important to note that the existing simulations of the GFDTD scheme employed analytical solutions to obtain function values at the points along the boundary; however, in simulations for which the analytical solution is unknown, theoretical approximations for values at points along the boundary are desperately needed. Hence, the objective of this dissertation research is to develop absorbing boundary conditions (ABCs) so that the GFDTD method can be used to solve the nonlinear Schrödinger equation when the analytical solution is unknown.
To create the ABCs for the nonlinear Schrödinger equation, we initially determine the associated EngquistMajda oneway wave equations and then proceed to develop a finite difference scheme for them. These ABCs are made to be adaptive using a windowed Fourier transform to estimate a value of the wavenumber of the carrier wave. These ABCs were tested using the nonlinear Schrödinger equation for 1D and 2D soliton propagation as well as Gaussian packet collision and dipole radiation. Results show that these ABCs perform well, but they have three key limitations. First, there are inherent reflections at the interface of the interior and boundary domains due to the different schemes used the two regions; second, to use the ABCs, one needs to estimate a value for the carrier wavenumber and poor estimates can cause even more reflection at the interface; and finally, the ABCs require different schemes in different regions of the boundary, and this domain decomposition makes the ABCs tedious both to develop and to implement.
To address these limitations for the FDTD method, we employ the fractionalorder derivative concept to unify the Schrödinger equation with its oneway wave equation over an interval where the fractional order is allowed to vary. Through careful construction of a variableorder fractional momentum operator, outgoing waves may enter the fractionalorder region with little to no reflection and, inside this region, any reflected portions of the wave will decay exponentially with time. The fractional momentum operator is then used to create a fractionalorder FDTD scheme. Importantly, this single scheme can be used for the entire computational domain, and the scheme smooths the abrupt transition between the FDTD method and the ABCs. Furthermore, the fractional FDTD scheme relaxes the precision needed for the estimated carrier wavenumber. This fractional FDTD scheme is tested for both the linear and nonlinear Schrödinger equations. Example cases include a 1D Gaussian packet scattering off of a potential, a 1D soliton propagating to the right, as well as 2D soliton propagation, and the collision of Gaussian packets. Results show that the fractional FDTD method outperforms the FDTD method with ABCs
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
Fourth order real space solver for the time-dependent Schr\"odinger equation with singular Coulomb potential
We present a novel numerical method and algorithm for the solution of the 3D
axially symmetric time-dependent Schr\"odinger equation in cylindrical
coordinates, involving singular Coulomb potential terms besides a smooth
time-dependent potential. We use fourth order finite difference real space
discretization, with special formulae for the arising Neumann and Robin
boundary conditions along the symmetry axis. Our propagation algorithm is based
on merging the method of the split-operator approximation of the exponential
operator with the implicit equations of second order cylindrical 2D
Crank-Nicolson scheme. We call this method hybrid splitting scheme because it
inherits both the speed of the split step finite difference schemes and the
robustness of the full Crank-Nicolson scheme. Based on a thorough error
analysis, we verified both the fourth order accuracy of the spatial
discretization in the optimal spatial step size range, and the fourth order
scaling with the time step in the case of proper high order expressions of the
split-operator. We demonstrate the performance and high accuracy of our hybrid
splitting scheme by simulating optical tunneling from a hydrogen atom due to a
few-cycle laser pulse with linear polarization
An overview on deep learning-based approximation methods for partial differential equations
It is one of the most challenging problems in applied mathematics to
approximatively solve high-dimensional partial differential equations (PDEs).
Recently, several deep learning-based approximation algorithms for attacking
this problem have been proposed and tested numerically on a number of examples
of high-dimensional PDEs. This has given rise to a lively field of research in
which deep learning-based methods and related Monte Carlo methods are applied
to the approximation of high-dimensional PDEs. In this article we offer an
introduction to this field of research, we review some of the main ideas of
deep learning-based approximation methods for PDEs, we revisit one of the
central mathematical results for deep neural network approximations for PDEs,
and we provide an overview of the recent literature in this area of research.Comment: 23 page
Effective sound absorbing boundary conditions for complex geometries
We analyze a system of equations that describes the propagation
of sound waves. We are interested in complex constructions along a
part of the boundary of the domain, for example constructions with small
chambers that are connected to the domain. We also allow that different
flow equations are used in the chambers, e.g., modelling a damping material.
In addition to the complex geometry, we assume that the viscosity
vanishes in the limit. The limiting system is given by wave equations,
we derive these equations and determine the effective boundary conditions.
The effective boundary conditions replace the large number of
small chambers. We provide examples for sound absorbing constructions
and their Dirichlet-to-Neumann boundary conditions
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