A Fully Discrete Truly Multidimensional Active Flux Method For The Two-Dimensional Euler Equations

TL;DR

This paper introduces a third-order accurate, fully discrete multidimensional Active Flux method for 2D Euler equations, featuring positivity-preserving limiters and effective boundary condition implementation, demonstrating accurate results on coarse grids.

Contribution

It develops a novel fully discrete multidimensional Active Flux method with positivity-preserving limiters for the 2D Euler equations, enhancing stability and accuracy.

Findings

Method achieves third-order accuracy.

Positivity of pressure and density is guaranteed.

Accurate results obtained on coarse computational grids.

Abstract

The Active Flux method is a finite volume method for hyperbolic conservation laws that uses both cell averages and point values as degrees of freedom. Several versions of such methods are currently under development. We focus on third order accurate, fully discrete Active Flux methods with compact stencil in space and time. These methods require exact or approximate evolution operators for the update of the point value degrees of freedom which are provided by the method of bicharacteristics. Here we propose new limiting strategies that guarantee positivity of pressure and density and furthermore discuss the implementation of reflecting boundary conditions. Numerical results show that the method leads to accurate approximates on coarse grids.