# The active flux scheme on Cartesian grids and its low Mach number limit

## TL;DR

This paper introduces an active flux finite volume scheme for 2D acoustic equations on Cartesian grids, demonstrating stability, accuracy in discontinuous solutions, and low Mach number compliance without additional fixes.

## Contribution

It extends the active flux scheme to two-dimensional Cartesian grids for acoustic equations, showing stability, accuracy, and low Mach number compliance.

## Key findings

- Stable simulation of discontinuous solutions.
- Low Mach number compliance without fixes.
- Effective implementation on 2D Cartesian grids.

## Abstract

Finite volume schemes for hyperbolic conservation laws require a numerical intercell flux. In one spatial dimension the numerical flux can be successfully obtained by solving (exactly or approximately) Riemann problems that are introduced at cell interfaces. This is more challenging in multiple spatial dimensions. The active flux scheme is a finite volume scheme that considers continuous reconstructions instead. The intercell flux is obtained using additional degrees of freedom distributed along the cell boundary. For their time evolution an exact evolution operator is employed, which naturally ensures the correct direction of information propagation and provides stability. This paper presents an implementation of active flux for the acoustic equations on two-dimensional Cartesian grids and demonstrates its ability to simulate discontinuous solutions with an explicit time stepping in a stable manner. Additionally, it is shown that the active flux scheme for linear acoustics is low Mach number compliant without the need for any fix.