A generalized Active Flux method of arbitrarily high order in two dimensions

TL;DR

This paper extends the semi-discrete Active Flux method to two dimensions, achieving arbitrarily high order accuracy by incorporating moments, and analyzes its stability and numerical performance for hyperbolic conservation laws.

Contribution

It introduces a high-order two-dimensional Active Flux method with moments, extending previous one-dimensional approaches and analyzing its stability and effectiveness.

Findings

The method achieves high-order accuracy in 2D.

Stability analysis for linear advection is provided.

Numerical examples demonstrate effectiveness for hyperbolic laws.

Abstract

The Active Flux method can be seen as an extended finite volume method. The degrees of freedom of this method are cell averages, as in finite volume methods, and in addition shared point values at the cell interfaces, giving rise to a globally continuous reconstruction. Its classical version was introduced as a one-stage fully discrete, third-order method. Recently, a semi-discrete version of the Active Flux method was presented with various extensions to arbitrarily high order in one space dimension. In this paper we extend the semi-discrete Active Flux method on two-dimensional Cartesian grids to arbitrarily high order, by including moments as additional degrees of freedom (hybrid finite element--finite volume method). The stability of this method is studied for linear advection. For a fully discrete version, using an explicit Runge-Kutta method, a CFL restriction is derived. We end by presenting numerical examples for hyperbolic conservation laws.