The aim of the project P-24157-N13 supported by the Austrian Science Fund (FWF) is to investigate the numerical solution of nonlinear Schrödinger equations. For the full discretization of initial value problems for these evolution equations we analyse the method of lines approach, which is suitable for problems where theoretical insight suggests a suitable space discretization and mesh. In this approach the evolutionary PDE is reduced to a (generally large) system of initial value problems for ordinary differential equations. For the resulting problems, splitting methods of high order will be investigated with respect to the structure of the local error and implications for the adaptive choice of time steps. Furthermore, new a priori and a posteriori error estimates based on the defect correction principle are put forward for the approximation of the matrix exponential function by splitting methods. This task occurs as a subproblem in the time integration of evolution equations in the context of the method of lines. A realization of this approach for nonlinear evolution equations will represent a nontrivial extension. Application of the project results to nonlinear single-particle Schrödinger equations associated with model reductions of the high-dimensional linear multi-particle Schrödinger equation like time-dependent density functional theory or the multi-configuration time-dependent Hartree-Fock method shall put the project results in an application-oriented context. Collaborators in the project are Winfried Auzinger (Vienna Universiyty of Technology), Gustaf Söderlind (University of Lund), and Mechthild Thalhammer (University of Innsbruck). So far, the project yielded the following articles and preprints and conference presentations. A movie illustrating the results of our research so far can be found here.

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last modification: Tue Oct 11 9:00 MET 2016