M. Ehrhardt (Bergische Universitaet Wuppertal) "A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schroedinger Equations" (joint work with X. Antoine, A. Arnold, C. Besse, A. Schaedle) Abstract: In this talk we discuss different techniques to solve numerically the time-dependent Schroedinger equation on unbounded domains. We present and compare several approaches to implement the classical transparent boundary condition into finite difference and finite element discretizations. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. Next, we show 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. In the second part we present some novel absorbing boundary conditions (ABCs) for modeling the solution of linear and nonlinear variable potentials one-dimensional stationary Schroedinger equations. Using pseudodifferential calculus and factorization theorems we construct a hierarchy of novel ABCs and generalize the well-known quantum transmitting boundary condition of Kirk and Lentner to the case of space-dependent potential. Moreover, we propose a rapidly converging iterative method based on finite elements suitable for computing scattering solutions and bound states. The accuracy of our new absorbing boundary conditions is investigated numerically for two different situations. The first problem is related to the computation of linear scattering problems. The second application concerns the computation of energies and ground-states for linear and nonlinear Schroedinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, our approach also offers the possibility to construct ABCs for higher dimensional problems. References: X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schaedle, "A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schroedinger Equations", Commun. Comput. Phys. Vol. 4, Number 4, (2008), 729-796. (open-access article) http://www.global-sci.com/openaccess/v4_729.pdf X. Antoine, C. Besse and P. Klein, Absorbing boundary conditions for the one-dimensional Schroedinger equation with an exterior repulsive potential, J. Comput. Phys. Vol. 228, (2009), 312-335. X. Antoine, C. Besse, M. Ehrhardt and P. Klein, "Modeling boundary conditions for solving stationary Schroedinger equations", Preprint 10/04, February 2010.