Abstract

In a series of unpublished papers, P.L. Roe and his students have started to develop a new third order method able to work on triangular unstructured meshes, using a continuous representation of data and no Riemann solvers. The degrees of freedom are the vertices of the elements and the mid points of the edges, as well as the average value of the conserved variables. The integration in time is done by a variant of the method of characteristics that allow to evaluate the flux at several sub-time steps. This explains the name “Active Flux”. This idea has then been developed by Roe and others, a very uncomplete list of references can be found at [T.A. Eyman & P.L. Roe 2011; C. Helzel, et al., JSC 2019; W. Barsukow, JSC 2021; E. Chudzik et al., JSC 2024; W. Barsukow, JSC 2019]. 

In this talk, I will make a personal review of this method using the material of [R. Abgrall, JSC 2019; R. Abgrall and W. Barsukow, ESIAM: M2AN 2023], show how to develop higher than third order schemes on triangles following [R. Abgrall, J. Lin, and Y. Liu]. Then I will explain the connection with the approximation tools provided by the Virtual Finite elements spaces and show several numerical results that confirms the stability and the accuracy of the methods. Some perspective will be drawn. 

This is a joint work with Yongle Liu (U. Zürich), Jianfang Lin (U. Zürich), W. Boscheri (U. Chambery, France) and W. Barsukow (U. Bordeaux, France).

Biography

Rémi Abgrall has been a professor of numerical analysis at the University of Zurich since 2014. He is a distinguished French applied mathematician, renowned for his significant contributions to computational fluid dynamics, numerical analysis of conservation laws, multiphase flow, and Hamilton–Jacobi equations. Since 2015, he has served as the editor-in-chief of the Journal of Computational Physics （JCP) and holds editorial positions in several other international scientific journals. In 2014, he was an invited speaker at the International Congress of Mathematics (ICM) in Seoul. In recognition of his groundbreaking work in developing numerical methods for conservation laws, particularly for multi-fluid flows and residual distribution schemes, he was elected as a SIAM Fellow in 2022.