Communications on Applied Mathematics and Computation >

2024 , Vol. 6 >Issue 3: 1860 - 1898

DOI: https://doi.org/10.1007/s42967-023-00321-6

ORIGINAL PAPERS

Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods

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  • 1 Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, Cologne 50931, Germany;
    2 Institute of Applied Mathematics (LS III), TU Dortmund University, Vogelpothsweg 87, Dortmund 44227, Germany;
    3 Center for Data and Simulation Science, University of Cologne, Weyertal 86-90, Cologne 50931, Germany

Received date: 2023-05-01

  Revised date: 2023-11-15

  Accepted date: 2023-09-17

  Online published: 2024-12-20

Supported by

The authors thank Dr. Hennes Hajduk for performing comparative studies with his code and giving a deeper insight into the subcell DG-MCL schemes he developed in [22] for high-order Bernstein finite elements. Moreover, the authors would like to thank Prof. Dr. Hendrik Ranocha for his support and advice during the implementation of FCT methods in Trixi.jl [57, 62, 63]. Gregor Gassner and Andrés M. RuedaRamírez acknowledge funding through the Klaus Tschira Stiftung via the project “HiFiLab”. Gregor Gassner further acknowledges funding by the German Research Foundation (DFG) under the grant number DFGFOR5409. Dmitri Kuzmin acknowledges DFG support under the grant number KU 1530/23-3. We furthermore thank the Regional Computing Center of the University of Cologne (RRZK) for providing computing time on the High Performance Computing (HPC) system ODIN, as well as for technical support.

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Abstract

We extend the monolithic convex limiting (MCL) methodology to nodal discontinuous Galerkin spectral-element methods (DGSEMS). 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 highorder 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. In addition, 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.

Key words： Structure-preserving schemes; Subcell flux limiting; Monolithic convex limiting (MCL); Discontinuous Galerkin spectral-element methods (DGSEMS); Legendre-Gauss-Lobatto (LGL) nodes

Cite this article

Andrés M. Rueda-Ramírez, Benjamin Bolm, Dmitri Kuzmin, Gregor J. Gassner . Monolithic Convex Limiting for Legendre-Gauss-Lobatto Discontinuous Galerkin Spectral-Element Methods[J]. Communications on Applied Mathematics and Computation, 2024 , 6(3) : 1860 -1898 . DOI: 10.1007/s42967-023-00321-6

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