Matthias Ehrhardt (WIAS Berlin)

Title: Discrete transparent boundary conditions for transient kp-Schr\"odinger equations
with application to quantum heterostructures

This talk is concerned with transparent boundary conditions (TBCs) 
for systems of Schr\"odinger-type equations, namely o the time-dependent kp-Schr\"odinger 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 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.