Frédérique Charles (Laboratoire Jacques-Louis Lions)
Particle methods for transport equations consist in pushing forward particles along the characteristic lines of the flow, and to describe then the transported density as a sum of weighted and smoothed particles. Conceptually simple, standard particle methods have the main drawback to produce noisy solutions or to require frequent remapping. In this talk we present two classes of particle methods which aim at improving the accuracy of the numerical approximations with a minimal amount of smoothing. The idea of the Linearly Transformed Particle method is to transform the shape functions of particles in order to follow the local variation of the flow. This method has been adapted and analyzed for the Vlasov- Poisson system and for a compressible aggregation equation. In both cases the error estimate is improved compared to classical particle methods, with the gain of a strong convergence of the numerical solution. However, for long remapping periods, shapes of particles could become to much stretched out. The second method solve this problem of locality by combining a backward semi-Lagrangian approach and local linearizations of the flow. The convergence properties are improved and validated by numerical experiments. This is a joint work with Martin Campos-Pinto (LJLL, UPMC).