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J. Melenk, A. Rieder:
"Runge-Kutta convolution quadrature and FEM-BEM coupling for the time dependent linear Schrödinger equation";
in: "ASC Report 13/2016", herausgegeben von: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2016, ISBN: 978-3-902627-09-4, S. 1 - 35.

We propose a numerical scheme to solve the time dependent linear Schrödinger equation. The discretization is carried out by combining a Runge-Kutta time-stepping scheme with a finite element iscretization in space. Since the Schr¨odinger equation is posed on the whole space
Rd we combine the interior finite element discretization with a convolution quadrature based boundary element discretization. In this paper we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.

http://www.asc.tuwien.ac.at/preprint/2016/asc13x2016.pdf