Dynamic low-rank approximations for kinetic equations: structure preservation and fluid limit (Lukas Einkemmer - University of Tübingen and University of Innsbruck)

30.10.2018 14:00 – 15:00

Many problems encountered in plasma physics require a kinetic description. The associated partial differential equations are posed in an up to six-dimensional phase space. Thus, a direct discretization of this phase space, often called the Eulerian approach, is extremely expensive from a computational point of view.

In this talk we propose a dynamical low-rank approximation for the Vlasov--Poisson and Vlasov--Maxwell equations. This approximation is derived by constraining the dynamics to a manifold of low-rank functions via a tangent space projection and by splitting this projection into the subprojections from which it is built. This reduces a time step for the six- (or four-) dimensional Vlasov equation to solving two systems of three- (or two-) dimensional advection equations. By a hierarchical dynamical low-rank approximation, a time step for the Vlasov equation can be further reduced to a set of six (or four) systems of one-dimensional advection equations.

The resulting systems of advection equations can then be solved by semi-Lagrangian or spectral methods. We highlight the favorable behavior of the proposed numerical method by performing numerical simulation for a number of problems. These simulation show that that the proposed algorithm is able to drastically reduce the required computational effort. Unfortunately, much of the physical structure inherent in the continuous problem is destroyed in the process. We will discuss an approach to restore some of the physical structure to the low-rank approximation, which is important even for relatively short times. In addition, we will discuss the underlying fluid limit.

Lieu

Room 623, Séminaire d'analyse numérique

entrée libre