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.

翻译：本文针对一维和二维带已知势函数源项的十矩高斯闭合方程，提出了新型高阶精度间断伽辽金（DG）格式。所提出的DG格式具备两方面理想特性：既能对已知静水平衡态达到平衡（WB），又能同时保持密度正性和各向异性压力张量的正定性。平衡性通过精确修改Harten-Lax-van Leer-contact（HLLC）通量中的解状态，并结合源项的适当重构与离散来实现。所提出的新型修改技术克服了各向异性效应带来的困难，在保持高阶精度的同时，确保修改后的解状态始终位于物理容许状态集内。通过利用容许状态集、HLLC通量与HLLC求解器的若干关键性质，以及[Wu & Shu, SIAM Review, 65: 1031-1073, 2023]中的几何拟线性化（GQL）方法（该方法最初由[Wu & Tang, M3AS, 27: 1871-1928, 2017]应用于分析相对论磁流体力学的容许状态集与物理约束保持格式），本文对WB DG格式进行了保正性分析，以处理压力张量非线性约束带来的困难。此外，所提出的WB DG格式满足单元平均值的弱正性，这表明可通过缩放限制器在特定关注点强制DG解多项式满足物理容许性。大量数值实验验证了该方法在平衡态保持、小扰动捕捉精度、低密度/低压问题鲁棒性以及光滑与间断解的高分辨率方面的有效性。