Extensions of Active Flux to arbitrary order of accuracy

TL;DR

This paper extends the Active Flux numerical method for hyperbolic conservation laws to arbitrary high order accuracy by increasing stencil size, point values, or degrees of freedom, while maintaining global continuity.

Contribution

It introduces multiple approaches to enhance Active Flux's order of accuracy, connecting it with finite volume, finite difference, and finite element methods.

Findings

Different extension strategies have distinct properties.

Extensions enable higher order accuracy while preserving continuity.

The methods relate Active Flux to classical numerical frameworks.

Abstract

Active Flux is a recently developed numerical method for hyperbolic conservation laws. Its classical degrees of freedom are cell averages and point values at cell interfaces. These latter are shared between adjacent cells, leading to a globally continuous reconstruction. The update of the point values includes upwinding, but without solving a Riemann Problem. The update of the cell average requires a flux quadrature at the cell interface, which can be immediately performed using the point values. This paper explores different extensions of Active Flux to arbitrarily high order of accuracy, while maintaining the idea of global continuity. We propose to either increase the stencil while keeping the same degrees of freedom, or to increase the number of point values, or to include higher moments as new degrees of freedom. These extensions have different properties, and reflect different views upon the relation of Active Flux to the families of Finite Volume, Finite Difference and Finite Element methods.