We present a general family of subcell limiting strategies to construct robust high-order accurate nodal discontinuous Galerkin (DG) schemes. The main strategy is to construct compatible low order finite volume (FV) type discretizations that allow for convex blending with the high-order variant with the goal of guaranteeing additional properties, such as bounds on physical quantities and/or guaranteed entropy dissipation. For an implementation of this main strategy, four main ingredients are identified that may be combined in a flexible manner: (i) a nodal high-order DG method on Legendre-Gauss-Lobatto nodes, (ii) a compatible robust subcell FV scheme, (iii) a convex combination strategy for the two schemes, which can be element-wise or subcell-wise, and (iv) a strategy to compute the convex blending factors, which can be either based on heuristic troubled-cell indicators, or using ideas from flux-corrected transport methods. By carefully designing the metric terms of the subcell FV method, the resulting methods can be used on unstructured curvilinear meshes, are locally conservative, can handle strong shocks efficiently while directly guaranteeing physical bounds on quantities such as density, pressure or entropy. We further show that it is possible to choose the four ingredients to recover existing methods such as a provably entropy dissipative subcell shock-capturing approach or a sparse invariant domain preserving approach. We test the versatility of the presented strategies and mix and match the four ingredients to solve challenging simulation setups, such as the KPP problem (a hyperbolic conservation law with non-convex flux function), turbulent and hypersonic Euler simulations, and MHD problems featuring shocks and turbulence.
翻译:我们提出了一个子细胞总体限制策略,以构建稳妥的高顺序准确交点不连续的 Galerkin (DG) 计划。主要策略是构建兼容的低顺序固定体积(FV) 类型分解(FV) 与高顺序变异结合的螺旋组合,目的是保证更多的属性,例如物理数量和/或保证的导流分解的界限。为了实施这一主策略,可以灵活地组合四个主要成分:(i) 在Tulturre-Gaus-Labatto 节点上,一个节点高顺序高调DG 方法,(ii) 一个兼容的低顺序固定体积体积分量(FVV) 子节量组合(FV) 机制, (ii) 兼容的低顺序稳定基体积分解(FV) 组合体积分解(FV) 组合体积(FVD) 组合体积(FV ) 的节流方法, 由此产生的方法可以用于不固定的流流法系, 以不固定的节流压法则(Weal rodeal) rodeal rodealeval) Procialevaleval) 。