Semi-discrete Active Flux as a Petrov-Galerkin method

Abstract: Active Flux (AF) is a recent numerical method for hyperbolic conservation laws, whose degrees of freedom are averages/moments and (shared) point values at cell interfaces. It has been noted previously in a heuristic fashion that it thus combines ideas from Finite Volume/Discontinuous Galerkin (DG) methods with a continuous approximation common in continuous Finite Element (CG) methods. This work shows that the semi-discrete Active Flux method on Cartesian meshes can be obtained from a variational formulation through a particular choice of (biorthogonal) test functions. These latter being discontinuous, the new formulation emphasizes the intermediate nature of AF between DG and CG.