Frédérique Charles[a]
Particle methods for transport equations: foundations and improved reconstruction methods
Pages: 333-373
Received: 20 July 2023
Accepted in revised form: 15 April 2025
Mathematics Subject Classification: 35F10, 65M25.
Keywords: Transport equations, characteristics, particle method, linearization.
Authors address:
[a]: Laboratoire Jean Kuntzmann, UMR CNRS 5224, Université Grenoble Alpes, Grenoble, France
Abstract: We present here two methods for improving the accuracy of the solution reconstruction in particle methods. First, we recall the mathematical theory of the transport equation, which is used in the following, and the principle of the smooth particle method. We then present the LTP method, where the shape functions of the particles are transformed according to a local expansion in order to follow the local variation of the flow. This method allows to obtain a strong convergence for the reconstructed solution without extensive overlapping of the particles. We then present a second method, the FBL method, which uses the backward Lagrangian representation of the solution and also the local expansions of the flow around the particles. The convergence analysis shows a theoretical gain of at least one order of convergence compared to the LTP method.