# A second order well-balanced finite volume scheme for Euler equations with gravity: general equation of state

We present a novel well-balanced second order Godunov-type finite volume scheme for compressible Euler equations with gravity. We consider general equation of state (EOS) for which explicit, analytical stationary solutions may not be available. Instead we compute a discrete stationary solution using a specific integration rule applied to the hydrostatic equations. The numerical scheme is designed to preserve this discrete stationary solution on any mesh. When applied to ideal gas model, the scheme exactly preserves the exact hydrostatic solution for isothermal and polytropic cases. As an example of general EOS we consider the ideal gas model together with radiation pressure. To study the accuracy of the solutions in computing perturbations around the stationary state, we solve the linearized Euler equations using a spectral collocation method and compare the solutions from the well-balanced non-linear finite volume scheme. These studies show that the well-balanced scheme gives accurate predictions of perturbations around the stationary solutions.

**People**: Praveen C, Jonas Berberich, Christian Klingenberg

**Status**: Ongoing work