We present a graph-based discretization method for solving hyperbolic systems of conservation laws using discontinuous finite elements. The method is based on the convex limiting technique technique introduced by Guermond et al. (SIAM J. Sci. Comput. 40, A3211--A3239, 2018). As such, these methods are mathematically guaranteed to be invariant-set preserving and to satisfy discrete pointwise entropy inequalities. In this paper we extend the theory for the specific case of discontinuous finite elements, incorporating the effect of boundary conditions into the formulation. From a practical point of view, the implementation of these methods is algebraic, meaning, that they operate directly on the stencil of the spatial discretization. This first paper in a sequence of two papers introduces and verifies essential building blocks for the convex limiting procedure using discontinuous Galerkin discretizations. In particular, we discuss a minimally stabilized high-order discontinuous Galerkin method that exhibits optimal convergence rates comparable to linear stabilization techniques for cell-based methods. In addition, we discuss a proper choice of local bounds for the convex limiting procedure. A follow-up contribution will focus on the high-performance implementation, benchmarking and verification of the method. We verify convergence rates on a sequence of one- and two-dimensional tests with differing regularity. In particular, we obtain optimal convergence rates for single rarefaction waves. We also propose a simple test in order to verify the implementation of boundary conditions and their convergence rates.

翻译：本文提出了一种基于图的非连续有限元离散方法，用于求解双曲守恒律系统。该方法基于Guermond等人（SIAM J. Sci. Comput. 40, A3211–A3239, 2018）提出的凸限制技术，从而在数学上保证不变集保持性及满足离散逐点熵不等式。我们针对非连续有限元的特殊情况扩展了理论，将边界条件的影响融入公式体系。从实践角度看，该方法具有代数可操作性，即直接作用在空间离散的模板上。作为两篇系列论文的第一篇，本文介绍并验证了基于非连续伽辽金离散的凸限制流程中的关键构建模块。具体而言，我们讨论了一种具有最小稳定化处理的高阶非连续伽辽金方法，该方法在单元基方法中展现出与线性稳定化技术相当的最优收敛速率。此外，我们还探讨了凸限制流程中局部边界的合理选择。后续论文将聚焦于该方法的并行高性能实现、基准测试及验证。我们通过一系列具有不同正则性的一维和二维数值测试验证了收敛速率，特别针对单一稀疏波获得了最优收敛阶。同时，我们设计了一个简单测试以验证边界条件实现及其收敛速率。