Well-Balanced Discontinuous Galerkin Methods for the Euler Equations Under Gravitational Fields

Euler equations under gravitational field admit hydrostatic equilibrium state where the flux produced by the pressure is exactly balanced by the gravitational source term. In this paper, we present well-balanced Runge-Kutta discontinuous Galerkin methods which can preserve the isothermal hydrostatic balance state exactly and maintain genuine high order accuracy for general solutions. To obtain the well-balanced property, we first reformulate the source term, and then approximate it in a way which mimics the discontinuous Galerkin approximation of the flux term. Extensive one- and two-dimensional simulations are performed to verify the properties of these schemes such as the exact preservation of the hydrostatic balance state, the ability to capture small perturbation of such state, and the genuine high order accuracy in smooth regions.