Time adaptive Zassenhaus splittings for the Schrödinger equation in the semiclassical regime

Abstract

Time dependent Schrödinger equations with conservative force field U commonly constitute a major challenge in the numerical approximation, especially when they are analysed in the semiclassical regime. Extremely high oscillations originate from the semiclassical parameter, and call for appropriate methods. We propose to employ a combination of asymptotic Zassenhaus splitting with $ Time adaptive Zassenhaus splittings 2 time adaptivity. While the former turns the disadvantage of the semiclassical parameter into an advantage, leading to highly efficient methods with low error constants, the latter enables to choose an optimal time step and to speed up the calculations when the oscillations subside. We support the results with numerical examples.

Optimal parameters for numerical solvers of PDEs

“…In the context of backward error analysis, the defect measures the discrepancy between the differential equation satisfied by the numerical solution and the original equation [26]. Defect based error estimates have been utilized widely in the development of time-adaptive methods for ordinary differential equations (ODEs) [12,21] and PDEs [3,4,6], but, to the best of our knowledge, these have not been employed for the estimation of optimal parameters. Unlike adaptive techniques for choosing time-steps, where the local error can be assumed to decrease monotonically with the time-step, in the proposed approach an optimization problem needs to be solved for finding the values of the parameters.…”