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

We consider the Euler equations with gravitational source term and propose a new well-balanced unstaggered central finite volume scheme, which can preserve the hydrostatic balance state exactly. The proposed scheme evolves a nonoscillatory numerical solution on a single grid, avoids the time consuming process of solving Riemann problems arising at the cell interfaces, and is second-order accurate both in time and space. Furthermore, the numerical scheme follows a well-balanced discretization that first discretizes the gravitational source term according to the discretization of the flux terms, and then mimics the surface gradient method and discretizes the density and energy according to the discretization of steady state density and energy functions, respectively. Finally, several numerical experiments demonstrating the performance of the well-balanced schemes in both one and two spatial dimensions are presented. The results indicate that the new scheme is accurate, simple, and robust.