On the equivalence of semi-discrete Active Flux and Discontinuous Galerkin methods and a comparison of their performance

TL;DR

This paper demonstrates the theoretical equivalence between Active Flux and Discontinuous Galerkin methods for linear problems, revealing their relationship, differences in efficiency, and explaining superconvergence phenomena through Radau polynomials.

Contribution

It establishes a formal mapping showing AF and DG are essentially the same method under certain conditions, clarifying their relationship and superconvergence behavior.

Findings

AF is more economical than DG for the same error level.

The methods are equivalent for linear problems in multiple dimensions.

Radau polynomials underpin the superconvergence of DG.

Abstract

The Active Flux (AF) method employs a globally continuous approximation, like continuous Finite Element methods. This is achieved through the placement of point values at cell interfaces which are shared between adjacent cells. With, on average, K+1 degrees of freedom per cell, Active Flux achieves a polynomial approximation of degree K+1, while the Discontinuous Galerkin (DG) method uses only polynomials of degree K, i.e. one degree less with the same number of degrees of freedom. Despite all the differences, in this paper we show, however, that for linear problems in one and several dimensions as well as -- in some sense -- for nonlinear ones, semi-discrete AF and DG are the same method. We identify a mapping between their respective degrees of freedom, upon which the updates of these degrees of freedom turn out to agree. On the one hand, AF therefore seems more economical then DG for a given value of the error, and we confirm this in numerical experiments. On the other hand, this is a way to understand superconvergence of DG in a natural way, and we show how Radau polynomials and their zeros appear in the mapping between DG and AF: In the Radau points, AF "shines through" as the background high-order scheme behind DG.