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

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.