This paper is concerned with transparent boundary conditions
(TBCs) for the time-dependent Schrödinger equation in one and two
dimensions. Discrete TBCs are introduced in the numerical simulations
of whole space problems in order to reduce the computational
domain to a finite region. Since the discrete TBC for the Schrödinger
equation includes a convolution w.r.t. time with a weakly decaying
kernel, its numerical evaluation becomes very costly for large-time
simulations.
As a remedy we construct approximate TBCs with a kernel having
the form of a finite sum-of-exponentials, which can be evaluated
in a very efficient recursion. We prove stability of the resulting initialboundary
value scheme, give error estimates for the considered approximation
of the boundary condition, and illustrate the efficiency of
the proposed method on several examples.
In the case of the equidistant discretization of the Airy differential equation (\discrete
Airy equation") the exact solution can be found explicitly. This fact is used
to derive a discrete transparent boundary condition (TBC) for a Schroedinger
equation with linear varying potential, which can be used in \parabolic equation"
simulations in (underwater) acoustics and for radar propagation in the troposphere.
We propose different strategies for the discrete TBC and show an efficient implementation.
Finally a stability proof for the resulting scheme is given. A numerical
example in the application to underwater acoustics shows the superiority of the new
discrete TBC.
We present a way to efficiently treat the well-known transparent boundary
conditions for the Schrödinger equation. Our approach is based on two ideas:
firstly, to derive a discrete transparent boundary condition (DTBC) based on the
Crank-Nicolson finite difference scheme for the governing equation. And, secondly,
to approximate the discrete convolution kernel of DTBC by sum-of-exponentials for
a rapid recursive calculation of the convolution. We illustrate the efficiency of the
proposed method on several examples.
This paper is concerned with the effficient implementation of transparent boundary conditions
(TBCs) for wide angle parabolic equations (WAPEs) assuming cylindrical symmetry.
In [1] a discrete TBC of convolution type was derived from the fully discretized whole?space
problem that is reflection?free and yields an unconditionally stable scheme. Since the discrete
TBC includes a convolution with respect to range with a weakly decaying kernel, its
numerical evaluation becomes very costly for long-range simulations.
As a remedy we construct new approximative transparent boundary conditions involving
exponential sums as an approximation to the convolution kernel. This special approximation
enables us to use a fast evaluation of the convolution type boundary condition.
This new approach was outlined in detail in [2] for the standard "parabolic" equation.
This work is concerned with transparent boundary conditions (TBCs) for systems of Schrödinger type equations, namely the time-dependent kp-Schrödinger equations. These TBCs
have to be constructed for the discrete scheme, in order to maintain stability and to avoid
numerical re
ections. The discrete transparent boundary conditions (DTBCs) are constructed
using the solution of the exterior problem with Laplace and Z-transformation respectively.
Hence we will analyse the numerical error caused by the inverse Z-transformation. Since
these DTBCs are non-local in time and thus very costly, we present approximate DTBCs,
that allow a fast calculation of the boundary terms.
In this work we derive an exact discrete artificial boundary condition
for the Crank-Nicolson scheme for solving the Black-Scholes
equation for the valuation of American options.
To ensure stability and to avoid any numerical reflections
we derive the artificial boundary
condition on a purely discrete level.
Since the exact discrete artificial boundary condition
includes a convolution with respect to time
with a weakly decaying kernel, its numerical evaluation
becomes very costly for large-time simulations.
As a remedy we construct approximate artificial boundary conditions
with a kernel having the form of a
finite sum-of-exponentials, which can be evaluated in a very
efficient recursion. We prove a simple stability criteria
for the approximated artificial boundary conditions.
Finally we illustrate the
efficiency of the proposed method on several examples
and compare it to previously obtained discretized artificial boundary conditions.
This work deals with the efficient numerical solution of
the two-dimensional one-way Helmholtz equation
posed on an unbounded domain.
In this case one has to introduce
artificial boundary conditions to confine the computational domain.
Here we construct with the Z-transformation
so-called discrete transparent
boundary conditions
for higher-order parabolic equations schemes.
These methods are Pade ``Parabolic'' approximations of
the one-way Helmholtz equation
and frequently used in integrated optics and (underwater) acoustics.
This paper is concerned with transparent boundary
conditions (TBCs) for the time-dependent Schrödinger equation
on a circular domain.
Discrete TBCs are introduced in the
numerical simulations of problems on unbounded domains in order to reduce
the computational domain to a finite region in order to make this problem feasible for numerical simulations.
The main focus of this article is on the
appropriate discretization of such
TBCs for the two-dimensional Schrödinger equation
in conjunction with a conservative Crank-Nicolson-type finite difference discretization.
The presented discrete TBCs yield an unconditionally stable
numerical scheme and are completely reflection-free at the boundary.
Furthermore we prove concisely the stability of the recurrence formulas used to
obtain the convolution coefficients of the new discrete TBC
for a spatially dependent potential.
This paper deals with the numerical solution of the time{dependent Schroedinger-
Poisson system in the spherically symmetric case. Since the problem is posed on an
unbounded domain one has to introduce artificial boundary conditions to confine
the computational domain. The main topic of this work is the construction of a
so-called discrete transparent boundary condition (TBC) for a Crank-Nicolsontype
predictor-corrector scheme for solving the Schroedinger-Poisson system. This
scheme has the property of mass and energy conservation exactly on the discrete
level. We propose different strategies for the discrete TBC and present an efficient
implementation. Finally, a numerical example illustrate the findings and shows the
comparison results between the different approaches.
In this work we construct and analyse transparent boundary conditions (TBCs)
for general systems of parabolic equations. These TBCs are constructed for the
fully discrete scheme (-method, finite differences), in order to maintain unconditional
stability of the scheme and to avoid numerical re
ections. The discrete
transparent boundary conditions (DTBCs) are discrete convolutions in time and
are constructed using the solution of the Z{transformed exterior problem. We will
analyse the numerical error of these convolution coefficients caused by the inverse
Z{transformation. Since the DTBCs are non{local in time and thus very costly to
evaluate, we present approximate DTBCs of a sum{of{exponentials form that allow
for a fast calculation of the boundary terms. Finally, we will use our approximate
DTBCs for an example of a
uid stochastic Petri net and present numerical results.
In this work we deal with the numerical solution of some problems of air pollution.
Since the problems are posed on unbounded domains we have to introduce
artificial boundaries to confine the computational region.
We construct and analyse (discrete) transparent boundary conditions
for an implicit difference scheme.
We discuss the concepts of positivity and monotonicity of
difference schemes and briefly consider these
properties of difference schemes for advection-diffusion equations
arising in problems of air (and water) pollution.
The efficiency and accuracy of our method is illustrated by an example.
This paper deals with the efficient numerical solution of the two-dimensional one
way Helmholtz equation posed on an unbounded domain. In this case one has to
introduce artificial boundary conditions to confine the computational domain. The
main topic of this work is the construction of so{called discrete transparent boundary
conditions for state-of-the-art parabolic equations methods, namely a split-step
discretization of the high{order parabolic approximation and the split-step Padle
algorithm of Collins. Finally, several numerical examples arising in optics and underwater
acoustics illustrate the efficiency and accuracy of our approach.
This work is concerned with transparent boundary conditions (TBCs) for systems of Schrödinger-type equations, namely
the time-dependent kp-Schrödinger equations. These TBCs are constructed for the fully discrete scheme (Crank-Nicolson,
finite differences), in order to maintain unconditional stability of the scheme and to avoid numerical reflections. The discrete
transparent boundary conditions (DTBCs) are discrete convolutions in time and are constructed using the Z-transformed
solution of the exterior problem. We will analyse the numerical error of these convolution coeffficients caused by the inverse
Z-transformation. Since the DTBCs are non-local in time and thus very costly to evaluate, we present approximate DTBCs
of a sum-of-exponentials form that allow for a fast calculation of the boundary terms.
We discuss the nonstandard problem of using the finite difference
method to solve numerically a partial differential equation posed on
an unbounded domain. We propose different strategies to construct
so-called discrete articial boundary conditions (ABCs) and present
an efficient implementation by the sum-of-exponential ansatz. The
derivation of the ABCs is based on the knowledge of the exact solution,
the construction of asymptotic solutions or the usage of a continued
fraction expansion to a second-order difference equation. Our approach
is explained by means of three different types of partial differential
equations arising in option pricing, in quantum mechanics and
in (underwater) acoustics. Finally, we conclude with an illustrating
numerical example from underwater acoustics showing the superiority
of our new approach.
Transparent boundary conditions (TBCs) are an important tool for the truncation of the compu-
tational domain in order to compute solutions on an unbounded domain. In this work we want
to show how the standard assumption of `compactly supported data' could be relaxed and derive
TBCs for the wide angle parabolic equation directly for the numerical scheme on the discrete level.
With this inhomogeneous TBCs it is not necessary that the starting field lies completely inside the
computational region. However, an increased computational effort must be accepted.
In this review article we discuss different techniques to solve numerically the
time-dependent Schrödinger equation on unbounded domains.
We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works.
We conclude with several numerical examples from
different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.