Semi-discrete Active Flux as a Petrov-Galerkin method: the case of one-dimensional and Cartesian grids

Active Flux (AF) is a 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 semi-discrete Active Flux methods 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.

Explicit constructions are given for the case of arbitrarily high-order Active Flux with additional moments in 1-d, and for the classical third-order Active Flux on two-dimensional Cartesian meshes.