Young Researchers Seminars, Maths Applications & Models (Cassini, Luise)

Fabio Cassini (Università di Verona)
Directional split exponential integrators: applications to ADR equations and related Turing patterns

Efficiently solving systems of evolutionary Advection-Diffusion-Reaction (ADR) equations is of paramount importance in practical applications. To this purpose, time marching schemes of exponential type received a lot of attention in the last years, thanks to the advances in effectively computing actions of the exponential-like matrix functions needed by these integrators. In this presentation, we show how to compute those actions for matrices that have d-dimensional Kronecker sum form. This structure appears, for instance, when discretizing in space the involved differential operators with standard finite differences. The proposed technique is based on a suitable directional splitting of the exponential-like matrix functions, for which an efficient evaluation through tensor-matrix products and Tucker operators can be realized. The implementation of such an approach is done by exploiting the high performance level 3 BLAS, which are available for basically any kind of modern computer architecture. The employment of the proposed technique allows for the effective numerical integration of many stiff problems from the applications using directional split exponential integrators. In particular, in this talk we present the results on two- and three-dimensional (systems of) ADR equations that lead to the so-called Turing patterns, in comparison with state-of-the-art techniques.

Martina Luise (Università di Trento)
Quantum vortex reconnections

This seminar is dedicated to the problem of reconnections of quantum vortices. First I present the Gross-Pitaevskii equation (GPE) that governs the dynamics and evolutions of (knotted) vortices. Then I introduce some topological invariant such helicity, linking number, writhing number and total twist that play a crucial role in the problem of reconnection. Finally I focus on vortex reconnection, a phenomenon that occurs when two (quantum) vortex lines approach each other and "reconnect", changing the global topology of the flow. I conclude the seminar with some numerical simulations.

Further information at this link