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:我们把单流线性二次曲线限制方法(MCL)推广到交点不连续的 Galerkin 光谱元件方法(DGSEM) 。 使用图例- Gaus- Lobatto(LGL) 的二次曲线底底线(LGL) 将DGSEM 空间问题分解为非线性双曲问题, 其特性大大简化了不易变域保护高分辨率计划的设计。 与许多其他连续和不连续的 Galerkin 方法变量相比, LGL 光谱操作器的一个特别优势是能够将自然分解成兼容的子流分解为兼容的子流分解。 代表中间状态的高级空间半分解(LGL) 矩形(LGL) 二次曲线底线底线(LGL) 底线分解(LGL- DGSEM 程序) 代表了不连续的空间中位空间半分流(L), 以保持这些状态的方式限制这些状态, 以及 Rungege- Kuttats seal ropeal rodeal rodeal roislal seal seal roisl roup for the demodal democal democal demodal develyal democal demodal develdal demodal develtal develtal ental develtal develtal develtal develtal develdal develdal develdal develdal develmental develds etal develds etal develments mutistictions etal develds ets etal develdal develdal etal etal develdal etal etal develdal develdal etal etal democal develdal etal se se sements etal etal etal et et et se se se comm etal comm et etal etal etal etal etal 。, 我们可进一步 </s>