Active Flux methods are high-order finite volume methods for hyperbolic conservation laws that combine cell averages with point values located on cell interfaces. Various versions of Active Flux methods are currently under development. In this talk, I focus on third-order accurate, fully discrete Active Flux methods with compact stencils in space and time. A crucial step of these methods is the evolution of the point values. To update these degrees of freedom, exact or approximate evolution operators are required. The construction of such operators is therefore a key component of fully discrete Active Flux schemes.

We focus on the construction of evolution operators for the linearised Euler equations. By introducing moving coordinates, the advective part of the system can be removed, reducing the problem to the acoustic equations. This reformulation allows us to employ both exact and approximate truly multidimensional evolution operators originally developed for acoustics. In this way, the construction of evolution operators for the linearised Euler equations is directly linked to their acoustic counterparts, and many of their properties can be transferred to the Euler setting.

The resulting evolution operators form the basis of fully discrete Active Flux methods for the compressible Euler equations. We compare Active Flux methods based on exact and approximate evolution operators and investigate how the choice of evolution operator influences the resulting numerical schemes. Numerical experiments illustrate the performance of these methods for different flow regimes, ranging from discontinuous solution structures with shock waves to vortex structures in the low-Mach-number regime.