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.
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.
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 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.
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.