In this work we present a framework for enforcing discrete maximum principles in discontinuous Galerkin (DG) discretizations. The developed schemes are applicable to scalar conservation laws as well as hyperbolic systems. Our methodology for limiting volume terms is similar to recently proposed methods for continuous Galerkin approximations, while DG flux terms require novel stabilization techniques. Piecewise Bernstein polynomials are employed as shape functions for the DG spaces, thus facilitating the use of very high order spatial approximations. We discuss the design of a new, provably invariant domain preserving DG scheme that is then extended by state-of-the-art subcell flux limiters to obtain a high-order bound preserving approximation. The limiting procedures can be formulated in the semi-discrete setting. Thus convergence to steady state solutions is not inhibited by the algorithm. We present numerical results for a variety of benchmark problems. Conservation laws considered in this study are linear and nonlinear scalar problems, as well as the Euler equations of gas dynamics and the shallow water system.

中文翻译：

---

双曲线守恒定律的不连续伽辽金离散中的单片凸限制

在这项工作中，我们提出了一个框架，用于在不连续伽辽金 (DG) 离散化中执行离散最大值原则。所开发的方案适用于标量守恒定律以及双曲线系统。我们用于限制体积项的方法类似于最近提出的用于连续伽辽金近似的方法，而 DG 通量项需要新颖的稳定技术。分段伯恩斯坦多项式被用作 DG 空间的形状函数，从而有助于使用非常高阶的空间近似。我们讨论了一种新的、可证明不变的域保持 DG 方案的设计，然后通过最先进的子单元通量限制器对其进行扩展以获得高阶边界保持近似。可以在半离散设置中制定限制程序。因此，算法不会阻止收敛到稳态解。我们提供了各种基准问题的数值结果。本研究中考虑的守恒定律是线性和非线性标量问题，以及气体动力学和浅水系统的欧拉方程。