The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.

中文翻译：

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ℙ1 标量双曲问题的不连续 Galerkin 离散化的熵稳定和保属性限制器

本文提出的方法弥合了非线性双曲线问题的熵稳定和保持正性的不连续伽辽金 (DG) 方法之间的差距。分别使用基于熵条件和离散最大值原理的通量限制器来执行熵稳定性属性以及可选地保留单元平均值的局部界限。由分段线性 DG 近似的（有限）梯度产生的熵使用 Rusanov 型熵粘度进行约束。熵稳定项的泰勒基表示表明，它以类似于斜率限制的方式惩罚解梯度，并且需要隐式处理以避免严重的时间步长限制。基于顶点的斜率限制器的可选应用将 DG 解决方案限制为单元平均值的局部最大值和最小值。对具有非线性和非凸通量函数的两个标量二维测试问题进行了数值研究。