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 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.
A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints
(2013)
We illustrate the solution of a mixed-integer stochastic nonlinear optimization problem in an application of power management. In this application, a coupled system consisting of a hydro power station and a wind farm is considered. The objective is to satisfy the local energy demand and sell any surplus energy on a spot market for a short time horizon. Generation of wind energy is assumed to be random, so that demand satisfaction is modeled by a joint probabilistic constraint taking into account the multivariate distribution. The turbine is forced to either operate between given positive limits or to be shut down. This introduces additional binary decisions. The numerical solution procedure is presented and results are illustrated.