A well-balanced finite volume scheme for the Euler equations with gravitation

Context. Many problems in astrophysics feature flows which are close to hydrostatic equilibrium. However, standard numerical schemes for compressible hydrodynamics may be deficient in approximating this stationary state, in which the pressure gradient is nearly balanced by gravitational forces. Aims. We aim to develop a second-order well-balanced scheme for the Euler equations. The scheme is designed to mimic a discrete version of the hydrostatic balance. Hence, it can resolve a discrete hydrostatic equilibrium exactly (up to machine precision) and propagate perturbations, on top of this equilibrium, very accurately. Methods. A local second-order hydrostatic equilibrium preserving pressure reconstruction is developed. Combined with a standard central gravitational source term discretization and numerical fluxes that resolve stationary contact discontinuities exactly, the wellbalanced property is achieved. Results. The resulting well-balanced scheme is robust and simple enough to be very easily implemented within any existing computer code solving time explicitly/implicitly the compressible hydrodynamics equations. We demonstrate the performance of the wellbalanced scheme for several astrophysically relevant applications: wave propagation in stellar atmospheres, a toy model for corecollapse supernovae, convection in carbon shell burning and a "realistic" proto-neutron star.