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 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.