Well-balanced Central Schemes for the One and Two-dimensional Euler Systems with Gravity

In this paper we develop a family of central schemes for the one and two-dimensional systems of Euler equations with gravitational source term. The proposed schemes are unstaggered, second-order, central finite volume schemes that avoid solving Riemann problems at the cell interfaces and avoid switching between an original and a staggered grid. The main feature of the schemes developed here is that they are capable of preserving any steady state of the Euler with gravity system up to machine accuracy by updating the numerical solution in terms of a relevant reference solution. The methodology proposed results in a well-balanced scheme capable of capturing any steady state. Our scheme is then implemented and used to solve classical problems from the recent literature.