Subcell limiting strategies for discontinuous Galerkin spectral element methods

Abstract

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.

26 pages, 11 figures

References in corpus (3)

• Athena: A New Code for Astrophysical MHD
• Adaptive numerical simulations with Trixi.jl: A case study of Julia for scientific computing
• Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems

Cited by in corpus (8)

• Toward Discretization-Consistent Closure Schemes for Large Eddy Simulation Using Reinforcement Learning
• GALÆXI: Solving complex compressible flows with high-order discontinuous Galerkin methods on accelerator-based systems
• Efficient Entropy-Stable Discontinuous Spectral-Element Methods Using Tensor-Product Summation-by-Parts Operators on Triangles and Tetrahedra
• Positivity-preserving entropy filtering for the ideal magnetohydrodynamics equations
• A flux-differencing formulation with Gauss nodes
• General relativistic force-free electrodynamics with a discontinuous Galerkin-finite difference hybrid method
• High order well-balanced Arbitrary-Lagrangian-Eulerian ADER discontinuous Galerkin schemes on general polygonal moving meshes
• Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration