We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral element methods (DGSEM). The use of Legendre-Gauss-Lobatto (LGL) quadrature endows collocated DGSEM space discretizations of nonlinear hyperbolic problems with properties that greatly simplify the design of invariant domain preserving high-resolution schemes. Compared to many other continuous and discontinuous Galerkin method variants, a particular advantage of the LGL spectral operator is the availability of a natural decomposition into a compatible subcell flux discretization. Representing a high-order spatial semi-discretization in terms of intermediate states, we perform flux limiting in a manner that keeps these states and the results of Runge-Kutta stages in convex invariant domains. Additionally, local bounds may be imposed on scalar quantities of interest. In contrast to limiting approaches based on predictor-corrector algorithms, our MCL procedure for LGL-DGSEM yields nonlinear flux approximations that are independent of the time-step size and can be further modified to enforce entropy stability. To demonstrate the robustness of MCL/DGSEM schemes for the compressible Euler equations, we run simulations for challenging setups featuring strong shocks, steep density gradients and vortex dominated flows.

翻译：我们将单片式凸限制方法推广至节点型间断伽辽金谱元方法。利用勒让德-高斯-洛巴托求积为非线性双曲问题的配点型间断伽辽金谱元空间离散赋予特性，极大简化了不变域保持的高分辨率格式设计。与其他连续及间断伽辽金方法变体相比，勒让德-高斯-洛巴托谱算子特有的优势在于其可实现自然分解为兼容的子单元通量离散格式。通过将高阶空间半离散表达为中间状态，我们以保持这些状态及龙格-库塔阶段结果位于凸不变域的方式进行通量限制。此外，可对感兴趣的标量量施加局部界约束。与基于预测-校正算法的限制方法不同，我们的勒让德-高斯-洛巴托-间断伽辽金谱元方法单片式凸限制流程生成不依赖于时间步长的非线性通量近似，并可进一步修改以确保熵稳定性。为证明单片式凸限制/间断伽辽金谱元格式对可压缩欧拉方程的鲁棒性，我们对包含强激波、陡峭密度梯度及涡主导流场的挑战性算例进行数值模拟。