A Second Order Well-Balanced Finite Volume Scheme for Euler Equations with Gravity

We present a novel well-balanced second order Godunov-type finite volume scheme for compressible Euler equations with gravity. The well-balanced property is achieved by a specific combination of source term discretization, hydrostatic reconstruction, and numerical flux that exactly resolves stationary contacts. The scheme is able to preserve isothermal and polytropic stationary solutions up to machine precision. It is applied on several examples using the numerical flux of Roe to demonstrate its well-balanced property and the improved resolution of small perturbations around the stationary solution.