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

TL;DR

The paper develops a generalized Active Flux method for hyperbolic conservation laws in two dimensions, achieving arbitrarily high order on Cartesian grids by coupling interior moments with shared edge point values in a hybrid finite element–finite volume framework. It introduces two reconstruction spaces (tensor-like and serendipity-like minimal DOFs) and provides a rigorous stability analysis, showing Gauss-edge point distributions yield stable semi-discrete schemes and deriving CFL bounds for fully discrete RK3 time stepping. Numerical experiments on linear advection, acoustics, and Euler equations demonstrate convergence at order N+1 and improved solution quality with higher order, validating the method's effectiveness on smooth problems. The work lays groundwork for limiting strategies and extensions to 3D, aiming at high-order, conservative, compact-stencil schemes suitable for complex hyperbolic systems.

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.

Figures (15)

• Figure 1: Degrees of freedom and reconstruction for a third-order Active Flux method on two-dimensional Cartesian grids.
• Figure 2: Superscripts specifying the degrees of freedom for a cell $C_{ij}$ for GenAF$(M_{})$ of fourth order (a generalized Active Flux method with minimal number of degrees of freedom, see Notation \ref{['Not_GenAFMMin']}). The circled point values mark the points belonging to a cell.
• Figure 3: Reference elements $C^\mathrm{ref}$ for GenAF$(M_{})$ of order $N+1$ (a generalized Active Flux method with minimal number of degrees of freedom, see Notation \ref{['Not_GenAFMMin']}).
• Figure 4: Shape functions for GenAF$(M_{})$ of 7th order.
• Figure 5: Example of finite difference formulas for GenAF$(M_{})$ of fifth order at locations of different degrees of freedom, marked by the additional circle (see \ref{['Eq_GenAFPVFDx']}, \ref{['Eq_GenAFPVFDy']}; coefficients rounded to two decimal points). Upwind direction: "$\rightarrow$".

Theorems & Definitions (2)

• Definition 1
• Definition 2