Adaptive semi-Lagrangian schemes for Vlasov equations

International audienceThis lecture presents a new class of adaptive semi-Lagrangian schemes - based on performing a semi-Lagrangian method on adaptive interpolation grids - in the context of solving Vlasov equations with underlying "smooth" flows, such as the one-dimensional Vlasov-Poisson system. After recalling the main features of the semi-Lagrangian method and its error analysis in a uniform setting, we describe two frameworks for implementing adaptive interpolations, namely multilevel meshes and interpolatory wavelets. For both discretizations, we introduce a notion of good adaptivity to a given function and show that it is preserved by a low-cost prediction algorithm which transports multilevel grids along any "smooth" flow. As a consequence, error estimates are established for the resulting predict and readapt schemes under the essential assumption that the flow underlying the transport equation, as well as its approximation, is a stable diffeomorphism. Some complexity results are stated in addition, together with a conjecture of the convergence rate for the overall adaptive scheme. As for the wavelet case, these results are new and also apply to high-order interpolation