Well-balanced finite difference weighted essentially non-oscillatory schemes for the Euler equations with static gravitational fields

Euler equations of compressible gas dynamics, coupled with a source term due to the gravitational fields, often appear in many interesting astrophysical and atmospheric applications. In this paper, we design high order finite difference weighted essentially non-oscillatory (WENO) methods for the Euler equations under static gravitation fields, which are well-balanced for known steady state solutions. We simplify the well-balanced WENO methods designed in Xing and Shu (2013) for the isothermal equilibrium, and then extend them to more general steady state solutions which include both isothermal and polytropic equilibria. One- and two-dimensional numerical examples are provided at the end to test the performance of the proposed WENO methods and verify these properties numerically.