Description
Abstract:
There is ample numerical evidente that splitting methods, when applied to the time integration of the (semi-discretized) Schrödinger equation, exhibit numerical resonances at specific values $h_r$ of the time step-size: for these values $h_r$ the errors in the solution and in the energy show a peak. E. Faou has analyzed in detail this phenomenon using backward error analysis techniques, and in particular has shown that resonances are closely related with the non-existence of a modified Hamiltonian.
In this talk, we introduce an alternative approach to explain the origin of these resonances and the non-existence of a modified equation when $h=h_r$ in the finite-dimensional setting. We also describe how the errors in the solution and in energy grow over time at resonance values of the time-step.
This is an ongoing work in collaboration with Sergio Blanes (Valencia), Ander Murua (San Sebasti\'an), and Mechthild Thalhammer (Innsbruck).
