  We develop, and implement in a Finite Volume environment, a density-based
approach for the Euler equations written in conservative form using density,
momentum, and total energy as variables. Under simplifying assumptions, these
equations are used to describe non-hydrostatic atmospheric flow. The
well-balancing of the approach is ensured by a local hydrostatic reconstruction
updated in runtime during the simulation to keep the numerical error under
control. To approximate the solution of the Riemann problem, we consider four
methods: Roe-Pike, HLLC, AUSM+-up and HLLC-AUSM. We assess our density-based
approach and compare the accuracy of these four approximated Riemann solvers
using two two classical benchmarks, namely the smooth rising thermal bubble and
the density current.
