Analysis of the multi-dimensional semi-discrete Active Flux method using the Fourier transform

TL;DR

This paper analyzes the multi-dimensional semi-discrete Active Flux method, focusing on its stability, diffusion, and stationarity properties for linear acoustics on Cartesian grids, using Fourier analysis.

Contribution

It extends the Active Flux method to multiple dimensions and provides a detailed Fourier-based analysis of its properties and stability.

Findings

The method is stationarity preserving for linear acoustics on Cartesian grids.

Analysis of numerical diffusion characteristics.

Stability properties are characterized through Fourier analysis.

Abstract

The degrees of freedom of Active Flux are cell averages and point values along the cell boundaries. These latter are shared between neighbouring cells, which gives rise to a globally continuous reconstruction. The semi-discrete Active Flux method uses its degrees of freedom to obtain Finite Difference approxi\-mations to the spatial derivatives which are used in the point value update. The averages are updated using a quadrature of the flux and making use of the point values as quadrature points. The integration in time employs standard Runge-Kutta methods. We show that this generalization of the Active Flux method in two and three spatial dimensions is stationarity preserving for linear acoustics on Cartesian grids, and present an analysis of numerical diffusion and stability.