High order well-balanced discontinuous Galerkin methods for Euler equations at isentropic equilibrium state under gravitational fields

Euler equations under gravitational fields often appear in some interesting astrophysical and atmospheric applications. The Euler equations are coupled with gravitational source term due to the gravity and admit hydrostatic equilibrium state where the flux produced by the pressure gradient is exactly balanced by the gravitational source term. In this paper, we construct high order discontinuous Galerkin methods for the Euler equations under gravitational fields, which are well-balanced for the isentropic type hydrostatic equilibrium state. To maintain the well-balanced property, we first reformulate the governing equations in an equivalent form. Then we propose a novel source term approximation based on a splitting algorithm as well as well-balanced numerical fluxes. Rigorous theoretical analysis and extensive numerical examples all suggest that the proposed methods maintain the hydrostatic equilibrium state up to the machine precision. Moreover, one- and two-dimensional simulations are performed to test the ability of the current methods to capture small perturbation of such equilibrium state, and the genuine high order accuracy in smooth regions.