  This paper presents a class of novel high-order accurate discontinuous
Galerkin (DG) schemes for the compressible Euler equations under gravitational
fields. A notable feature of these schemes is that they are well-balanced for a
general hydrostatic equilibrium state, and at the same time, provably preserve
the positivity of density and pressure. In order to achieve the well-balanced
and positivity-preserving properties simultaneously, a novel DG spatial
discretization is carefully designed with suitable source term reformulation
and a properly modified Harten-Lax-van Leer contact (HLLC) flux. Based on some
technical decompositions as well as several key properties of the admissible
states and HLLC flux, rigorous positivity-preserving analyses are carried out.
It is proven that the resulting well-balanced DG schemes, coupled with strong
stability preserving time discretizations, satisfy a weak positivity property,
which implies that one can apply a simple existing limiter to effectively
enforce the positivity-preserving property, without losing high-order accuracy
and conservation. The proposed methods and analyses are applicable to the Euler
system with general equation of state. Extensive one- and two-dimensional
numerical tests demonstrate the desired properties of these schemes, including
the exact preservation of the equilibrium state, the ability to capture small
perturbation of such state, the robustness for solving problems involving low
density and/or low pressure, and good resolution for smooth and discontinuous
solutions.
