Structure-Preserving Methods for the Euler Equations

In the last decade, there has been growing interest in developing new dynamical cores for climate and weather simulations based on the Discontinuous Galerkin (DG) approach. To achieve accuracy, efficiency, and stability, various numerical formulations of the Euler equations are currently being explored.

In this talk, we present novel structure-preserving methods for different formulations of the Euler equations. These include models based on different thermodynamic variables, such as potential temperature, internal energy, or total energy, as well as the Exner pressure formulation and the vector-invariant form. These methods are developed within the flux-differencing DG framework, with a specific focus on the efficient implementation of split forms for both conservative and non-conservative terms.

The implementation is carried out entirely in Julia and is integrated within the Trixi.jl and CGDycore.jl ecosystems. We will discuss the different architectural approaches used in these packages and showcase numerical results, specifically focusing on the baroclinic instability test case to compare the different formulations.