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

Wasilij Barsukow, Christian Klingenberg, Simon Krotsch

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.

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

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.
Paper Structure (20 sections, 7 theorems, 111 equations, 7 figures, 5 tables)

This paper contains 20 sections, 7 theorems, 111 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Cast
  3. Active Flux
  4. One spatial dimension
  5. Two spatial dimensions
  6. Radau polynomials
  7. Discontinuous Galerkin
  8. Equivalence between DG and Active Flux and superconvergence of DG
  9. One-dimensional linear problems
  10. One-dimensional nonlinear problems
  11. Multi-dimensional linear problems on Cartesian grids
  12. A special choice of degrees of freedom for AF
  13. The equivalence of DG and AF
  14. Performance Comparison
  15. Theoretical Expectations
  16. ...and 5 more sections

Key Result

Theorem 3.4

The (broken) DG approximation $q_i \in P^K$ augmented by the two Radau polynomials is the (globally continuous) AF approximation $Q_i\in P^{K+1}$ as it was introduced in Section ssec:af1d.

Figures (7)

  • Figure 1: Left: Radau polynomials $R^{K+1}_\text{L}$ for values $K \in \{ 1,\dots,5 \}$, with $\Delta x = 1$. Right: Overview of the equivalent DG and AF methods and their degrees of freedom.
  • Figure 2: Left: Classical degrees of freedom of Active Flux on Cartesian grids (third-order accuracy). Right: The modified degrees of freedom proposed here. The edge midpoints are replaced by one-dimensional averages along the edge.
  • Figure 3: The total runtime of the Active Flux and Discontinuous Galerkin methods of differnt order with respect to the number of cells.
  • Figure 4: The total runtime of the Active Flux and Discontinuous Galerkin methods of differnt with respect to the error $\mathcal{E}_\mathrm{dofs}$.
  • Figure 5: The average runtime per time step and the total runtime with respect to order of the Active Flux and Discontinuous Galerkin methods.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • Remark 3.7
  • Remark 3.8
  • ...and 14 more