  This paper proposes novel high-order accurate discontinuous Galerkin (DG)
schemes for the one- and two-dimensional ten-moment Gaussian closure equations
with source terms defined by a known potential function. Our DG schemes exhibit
the desirable capability of being well-balanced (WB) for a known hydrostatic
equilibrium state while simultaneously preserving positive density and
positive-definite anisotropic pressure tensor. The well-balancedness is built
on carefully modifying the solution states in the Harten-Lax-van Leer-contact
(HLLC) flux, and appropriate reformulation and discretization of the source
terms. Our novel modification technique overcomes the difficulties posed by the
anisotropic effects, maintains the high-order accuracy, and ensures that the
modified solution state remains within the physically admissible state set.
Positivity-preserving analyses of our WB DG schemes are conducted by using
several key properties of the admissible state set, the HLLC flux and the HLLC
solver, as well as the geometric quasilinearization (GQL) approach in [Wu &
Shu, SIAM Review, 65: 1031-1073, 2023], which was originally applied to analyze
the admissible state set and physical-constraints-preserving schemes for the
relativistic magnetohydrodynamics in [Wu & Tang, M3AS, 27: 1871-1928, 2017], to
address the difficulties arising from the nonlinear constraints on pressure
tensor. Moreover, the proposed WB DG schemes satisfy the weak positivity for
the cell averages, implying the use of a scaling limiter to enforce the
physical admissibility of the DG solution polynomials at certain points of
interest. Extensive numerical experiments are conducted to validate the
preservation of equilibrium states, accuracy in capturing small perturbations
to such states, robustness in solving problems involving low density or low
pressure, and high resolution for both smooth and discontinuous solutions.
