Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows

We develop structure-preserving numerical methods for the compressible Euler equations, employing potential temperature as a prognostic variable. We construct three numerical fluxes designed to ensure the conservation of entropy and total energy within the discontinuous Galerkin framework on general curvilinear meshes. Furthermore, we introduce a generalization for the kinetic energy preservation property and total energy conservation in the presence of a gravitational potential term. To this end, we adopt a flux-differencing approach for the discretization of the source term, treated as non-conservative product. We present well-balanced schemes for different constant background states for both formulations (total energy and potential temperature) on curvilinear meshes. Finally, we validate the methods by comparing the potential temperature formulation with the traditional Euler equations formulation across a range of classical atmospheric scenarios.