Active flux for triangular meshes for compressible flows problems

In this article, we show how to construct a numerical method for solving hyperbolic problems, whether linear or non-linear, using a continuous representation of the variables and their mean value in each triangular element.This type of approach has already been introduced by Roe, and others, in the multi-dimensional framework under the name of Active Flux, see [11,12,10,16,7].Here, the presentation is more general and follows [4,6].Various examples show the good behavior of the method in both linear and nonlinear cases, including non-convex problems.The expected order of precision is obtained in both the linear and non-linear cases.This work represents a step towards the development of methods in the spirit of virtual finite elements for linear or non-linear hyperbolic problems, including the case where the solution is not regular.