Well-Balanced Central-Upwind Schemes for the Euler Equations with Gravitation

In this paper, we develop a second-order well-balanced central-upwind scheme for the Euler equations of gas dynamics with gravitation. The proposed scheme is capable of exactly preserving steady-state solutions expressed in terms of a nonlocal equilibrium variable. A crucial step in the construction of the second-order scheme is a well-balanced piecewise linear reconstruction of equilibrium variables, which is combined with a well-balanced evolution in time, achieved by reducing the amount of numerical viscosity (present at the central-upwind scheme) in the areas where the flow is at (near) steady-state regime. We show the performance of our newly developed central-upwind scheme and demonstrate importance of perfect balance between the fluxes and gravitational forces on a number of oneand two-dimensional examples.