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 to update the point value degrees of freedom. In our method, these operators are provided by the method of bicharacteristics, which solves a local linearization of the nonlinear hyperbolic problem. We analyse the linearization error and propose a new version of a fully discrete active flux method, which is third order accurate for smooth problems. Furthermore, we propose limiting strategies for our fully discrete active flux method applied to the Euler equations of gas dynamics that guarantee positivity of density and pressure. Finally, we discuss the implementation of boundary conditions for these fully discrete active flux methods. Numerical results confirm that the method provides accurate results even when using coarse grids.