Conserving physical invariants for low-rank approximations

Many problems encountered in plasma physics require a kinetic description. Such problems are posed in an up to six-dimensional phase space. This makes their numerical treatment extremely expensive. Here we propose a dynamical low-rank approximation for the Vlasov equation. This approximation is derived by constraining the dynamics to a manifold of low-rank functions. This reduces a time step for the six-dimensional Vlasov equation to solving lower-dimensional advection equations. While this method is very efficient for a range of problems, the physical structure of the underlying partial differential equations is completely destroyed by the algorithm. In this talk, we will discuss some ideas how we can conserve physical invariants within a low-rank approach.