Semi-discrete Active Flux as a Petrov-Galerkin method

TL;DR

This paper demonstrates that the semi-discrete Active Flux method for hyperbolic conservation laws can be derived from a variational formulation, highlighting its intermediate nature between Discontinuous Galerkin and Continuous Galerkin methods.

Contribution

It establishes a variational formulation for Active Flux, clarifying its theoretical foundation and its relation to DG and CG methods.

Findings

Active Flux can be derived from a variational principle.

The formulation uses biorthogonal test functions.

Highlights AF's intermediate position between DG and CG.

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.