Diploma and Master Theses (authored and supervised):
"Exponential integrators for time-dependent multi-particle Schrödinger equations";
Supervisor: W. Auzinger;
Institut für Analysis und Scientific Computing,
final examination: 2020-04-21.
We compare diﬀerent time stepping methods for the multi-conﬁguration time-dependent Hartree-Fock (MCTDHF) method for the Schršodinger equation. Especially we focus on exponential integrators, where the diﬀerential equation is ﬁrst transformed using the variation-of-constants formula or via the Lawson transformation and then solved numerically. First we compare the methods on a cubic Schršodinger equation with an exact solution to verify the expected convergence behaviour of our implementation and to numerically compare properties such as the error constant. Then we use two model problems - a helium atom and a quantum dot that we irradiate by an external potential (laser pulse) - which we discretize using the MCTDHF forfurtherevaluation. Ontheseproblemswewilladditionallyevaluateadaptivemultistep methods (again using an exponential transformation) and observe the change in the size of the time step. We ﬁnd that although the exponential one-step methods (using either transformation) provide excellent stability results, the Adams-Lawson multi-step methods with a predictor-corrector step are the most eﬃcient methods due to the ability to increase the convergence order arbitrarily at virtually no extra computational cost. The eﬃciency is further increased using time step adaptivity, where we observe that the time step prediction reacts to local extrema of the external potential which seem to pose a stability requirement for the methods.
Created from the Publication Database of the Vienna University of Technology.