Well-balanced Discontinuous Galerkin Finite Element Method For Euler Equations In Gravity Field

Euler equations under gravitational fields admit hydrostatic equilibrium state where the flux produced by the pressure gradient is exactly balanced by the gravitational source term.In this paper,we present WB discontinuous Galerkin methods.The present methods can preserve the isentropic equilibrium state exactly and maintain genuine high order accuracy for general solutions.To obtain the WB property,with the help of the isentropic equilibrium state solutions,we first reformulate the governing equations in an equivalent form,and then propose a novel source term approximation as well as WB numerical fluxes.Rigorous theoretical analysis as well as extensive numerical examples all suggest that the present methods maintain the WB property.Moreover,one-andtwo-dimensional simulations are performed to test the ability to capture small perturbation of such isentropic equilibrium state,and the genuine high order accuracy in smooth regions.