Subcell limiting strategies for discontinuous Galerkin spectral element methods (2202.00576v2)

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.

• The paper introduces a unified framework that combines high-order DG methods with compatible subcell finite-volume strategies to enhance shock-capturing accuracy.
• It employs a convex blending strategy with dynamically computed coefficients to seamlessly transition between high-order and low-order discretizations.
• Numerical experiments on Kelvin-Helmholtz instability and astrophysical jets demonstrate reduced dissipation and improved solution fidelity.

Subcell Limiting Strategies for Discontinuous Galerkin Spectral Element Methods

The paper "Subcell Limiting Strategies for Discontinuous Galerkin Spectral Element Methods" by Rueda-Ramírez et al. presents a comprehensive framework for enhancing the robustness of nodal Discontinuous Galerkin Spectral Element Methods (DGSEM) through a family of subcell limiting strategies. These strategies are notably aligned with high-order accuracy requirements while ensuring numerical stability in the presence of strong shocks and turbulence prevalent in solving hyperbolic conservation laws.

Methodological Contributions

The authors identify a flexible design framework for constructing robust high-order DG schemes by integrating four main methodological ingredients:

1. High-Order DG Method: The framework adopts a nodal DG approach based on Legendre-Gauss-Lobatto nodes, enabling high-order polynomial approximations.
2. Compatible Subcell Finite Volume (FV) Method: A low-order FV discretization complements the high-order scheme. This FV method is reconstructed on subcell meshes compatible with the high-order nodal points, ensuring conservation properties and robustness to solution discontinuities.
3. Convex Blending Strategy: The paper examines both element-wise and subcell-wise convex combinations between the high-order DG and low-order FV schemes. The convex blending strategy enhances the flexibility of the numerical method in adapting to varying solution smoothness across the computational domain.
4. Computation of Blending Coefficients: Blending coefficients are computed using either a priori or a posteriori strategies, with the latter derived from concepts in flux-corrected transport methods. These coefficients dynamically adjust the degree of blending based on heuristic troubled-cell indicators or rigorous mathematical constraints to ensure physical plausibility, such as positivity preservation and entropy conditions.

Numerical Results and Observations

The versatility and efficacy of the subcell limiting strategies are validated through several challenging test problems. Noteworthy among these is the application to the Kelvin-Helmholtz instability and high Mach number astrophysical jets, where the proposed methods successfully resolve complex flow features with minimized numerical dissipation. The framework is demonstrated to recover existing methods like entropy-stable discretizations and invariant domain preserving approaches, showcasing its adaptability.

Numerical experiments reveal that subcell-wise blending generally achieves lower dissipation compared to element-wise approaches, crucially maintaining the fidelity of high-order accuracy. The approach thus promises robust and efficient handling of hyperbolic systems, including adaptable strategies for turbulence and shock waves in Magnetohydrodynamics (MHD).

Implications and Future Directions

The implications of this research are significant both theoretically and practically. The flexibility and robustness enhancements introduced by this framework can facilitate the development of more efficient solvers for problems in fluid dynamics and beyond, particularly where sharp gradients or shocks are involved. Moreover, by offering a systematic approach to integrate various subcell limiters and blend them effectively, this paper paves the way for future research to explore innovative combinations of methodologies tailored for specific applications or novel classes of numerical problems.

In summary, this paper presents a unified and adaptable framework for implementing subcell limiting strategies in DGSEM, providing opportunities for enhanced robustness and accuracy in the numerical solution of complex hyperbolic PDEs. Future research may focus on further refining the blending strategies, exploring more sophisticated FV reconstructions and numerical flux functions to advance the state-of-the-art in DG spectral element methods.