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 the 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, as proposed by Abgrall in the context of nodal finite element approximations. We cast his algebraic entropy fix into a form suitable for arbitrary polynomial bases and, in particular, for modal DG approaches. 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 semi-implicit treatment to achieve the desired effect. The implicit Taylor basis version of the Rusanov entropy fix preserves the sparsity pattern of the element mass matrix. Hence, no linear systems need to be solved if the Taylor basis is orthogonal and an explicit treatment of the remaining terms is adopted. The optional application of a vertex-based slope limiter constrains the piecewise-linear DG solution to be bounded by local maxima and minima of the cell averages. The combination of entropy stabilization with flux and slope limiting leads to constrained approximations that possess all desired properties. Numerical studies of the new limiting techniques and entropy correction procedures are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.

中文翻译：

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非线性双曲方程不连续伽辽金离散的熵稳定和性质保持限制器

本文提出的方法弥补了非线性双曲线问题的熵稳定方法和保正性不连续伽辽金 (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强制执行熵稳定性属性和（可选地）保持单元平均值的局部边界。正如 Abgrall 在节点有限元近似的背景下所提出的那样，分段线性 DG 近似的（有限）梯度产生的熵受到 Rusanov 型熵粘度的约束。我们将他的代数熵固定转换为适用于任意多项式基的形式，尤其适用于模态 DG 方法。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，并且需要半隐式处理才能达到预期效果。Rusanov 熵修正的隐式泰勒基版本保留了元素质量矩阵的稀疏模式。因此，如果泰勒基是正交的并且对其余项采用显式处理，则不需要求解线性系统。基于顶点的斜率限制器的可选应用将分段线性 DG 解决方案限制为以单元平均值的局部最大值和最小值为界。熵稳定与通量和斜率限制的结合导致具有所有所需属性的约束近似。