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

doi:10.1007/s42967-023-00321-6

Abstract

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

CLC Number:

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.

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References

1. Anderson, R., Dobrev, V., Kolev, T., Kuzmin, D., Quezada de Luna, M., Rieben, R., Tomov, V.: Highorder local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation. J. Comput. Phys. 334, 102–124 (2017)
2. Bacigaluppi, P., Abgrall, R., Tokareva, S.: “A posteriori” limited high order and robust schemes for transient simulations of fluid flows in gas dynamics. J. Comput. Phys. 476, 111898 (2023)
3. Barth, T.J.: Numerical methods for gasdynamic systems on unstructured meshes. In: An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, pp. 195–285. Springer, Berlin (1999)
4. Chandrashekar, P.: Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations. Commun. Comput. Phys. 14(5), 1252–1286 (2013)
5. Cheng, J., Shu, C.-W.: Positivity-preserving Lagrangian scheme for multi-material compressible flow. J.Comput. Phys. 257, 143–168 (2014)
6. Cotter, C.J., Kuzmin, D.: Embedded discontinuous Galerkin transport schemes with localised limiters. J.Comput. Phys. 311, 363–373 (2016)
7. Diot, S., Clain, S., Loubère, R.: Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Comput. Fluids 64, 43–63(2012)
8. Dobrev, V., Kolev, T., Kuzmin, D., Rieben, R., Tomov, V.: Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations. J. Comput. Phys. 356, 372–390 (2018)
9. Dumbser, M., Zanotti, O., Loubère, R., Diot, S.: A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J. Comput. Phys. 278, 47–75 (2014)
10. Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252, 518–557 (2013)
11. Fryxell, B., Olson, K., Ricker, P., Timmes, F., Zingale, M., Lamb, D., MacNeice, P., Rosner, R., Truran, J.,Tufo, H.: Flash: an adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes.Astrophys. J. Suppl. Ser. 131(1), 273 (2000)
12. Galbraith, M., Murman, S., Kim, C., Persson, P., Fidkowski, K., Glasby, R., Hillewaert, K., Ahrabi, B.:5th International Workshop on High-Order CFD Methods. http://how5.cenaero.be. AIAA Sci. Technol. Forum Expos. (2017)
13. Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), 1233–1253 (2013)
14. Gassner, G.J., Winters, A.R., Kopriva, D.A.: Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 327, 39–66 (2016)
15. Guermond, J.-L., Nazarov, M., Popov, B., Tomas, I.: Second-order invariant domain preserving approximation of the Euler equations using convex limiting. SIAM J. Sci. Comput. 40(5), 3211–3239 (2018)
16. Guermond, J.-L., Pasquetti, R., Popov, B.: Entropy viscosity method for nonlinear conservation laws. J.Comput. Phys. 230(11), 4248–4267 (2011)
17. Guermond, J.-L., Popov, B.: Invariant domains and first-order continuous finite element approximation for hyperbolic systems. SIAM J. Numer. Anal. 54(4), 2466–2489 (2016)
18. Guermond, J.-L., Popov, B.: Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations. J. Comput. Phys. 321, 908–926 (2016)
19. Guermond, J.-L., Popov, B.: Invariant domains and second-order continuous finite element approximation for scalar conservation equations. SIAM J. Numer. Anal. 55(6), 3120–3146 (2017)
20. Guermond, J.-L., Popov, B., Tomas, I.: Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems. Comput. Methods Appl. Mech. Eng. 347, 143–175 (2019)
21. Ha, Y., Gardner, C.L., Gelb, A., Shu, C.-W.: Numerical simulation of high Mach number astrophysical jets with radiative cooling. J. Sci. Comput. 24(1), 29–44 (2005)
22. Hajduk, H.: Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws. Comput. Math. Appl. 87, 120–138 (2021)
23. Hajduk, H., Kuzmin, D., Kolev, T., Abgrall, R.: Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations. Comput. Methods Appl. Mech. Eng. 359, 112658 (2020)
24. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)
25. Hennemann, S., Rueda-Ramírez, A.M., Hindenlang, F.J., Gassner, G.J.: A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations. J. Comput. Phys. 426, 109935 (2021)
26. Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput. Phys. 228(15), 5410–5436 (2009)
27. Johnson, C., Szepessy, A., Hansbo, P.: On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws. Math. Comput. 54(189), 107–129 (1990)
28. Krivodonova, L., Xin, J., Remacle, J.-F., Chevaugeon, N., Flaherty, J.E.: Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48(3/4), 323–338 (2004)
29. Kuzmin, D.: Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws. Comput. Methods Appl. Mech. Eng. 361, 112804 (2020)
30. Kuzmin, D., Hajduk, H., Rupp, A.: Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems. Comput. Methods Appl. Mech. Eng. 389, 114428 (2022)
31. Kuzmin, D., Klyushnev, N.: Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations. J. Comput. Phys. 407, 109230 (2020)
32. Kuzmin, D., Löhner, R., Turek, S.: Flux-Corrected Transport: Principles, Algorithms, and Applications, 2nd edn. Springer, Dordrecht (2012)
33. Kuzmin, D., Möller, M., Shadid, J.N., Shashkov, M.: Failsafe flux limiting and constrained data projections for equations of gas dynamics. J. Comput. Phys. 229(23), 8766–8779 (2010)
34. Kuzmin, D., Quezada de Luna, M.: Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws. J. Comput. Phys. 411, 109411 (2020)
35. Kuzmin, D., Quezada de Luna, M.: Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws. Comput. Fluids 213, 104742 (2020)
36. Kuzmin, D., Quezada de Luna, M., Ketcheson, D.I., Grüll, J.: Bound-preserving convex limiting for high-order Runge-Kutta time discretizations of hyperbolic conservation laws. J. Sci. Comput. 91, 21 (2022)
37. Kuzmin, D., Vedral, J.: Dissipation-based WENO stabilization of high-order finite element methods for scalar conservation laws. arXiv:2212.14224 (2022)
38. Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13(2), 217–237 (1960)
39. Lin, Y., Chan, J., Thomas, I.: A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations. J. Comput. Phys. 475, 111850 (2023)
40. Liu, Y., Lu, J., Shu, C.-W.: An essentially oscillation-free discontinuous Galerkin method for hyperbolic systems. SIAM J. Sci. Comput. 44(1), A230–A259 (2022)
41. Lohmann, C., Kuzmin, D.: Synchronized flux limiting for gas dynamics variables. J. Comput. Phys. 326, 973–990 (2016)
42. Lohmann, C., Kuzmin, D., Shadid, J.N., Mabuza, S.: Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements. J. Comput. Phys. 344, 151–186 (2017)
43. Löhner, R.: Applied Computational Fluid Dynamics Techniques: an Introduction Based on Finite Element Methods, 2nd edn. Wiley, Chichester (2008)
44. Lv, Y., See, Y.C., Ihme, M.: An entropy-residual shock detector for solving conservation laws using high-order discontinuous Galerkin methods. J. Comput. Phys. 322, 448–472 (2016)
45. Maier, M., Kronbichler, M.: Efficient parallel 3D computation of the compressible Euler equations with an invariant-domain preserving second-order finite-element scheme. ACM Trans. Parall. Comput. 8(3), 1–30 (2021)
46. Markert, J., Gassner, G., Walch, S.: A sub-element adaptive shock capturing approach for discontinuous Galerkin methods. Commun. Appl. Math. Comput. 5, 679–721 (2023)
47. Mateo-Gabín, A., Rueda-Ramírez, A.M., Valero, E., Rubio, G.: Entropy-stable flux-differencing formulation with Gauss nodes for the DGSEM. arXiv:2211.05066 (2022)
48. Moe, S.A., Rossmanith, J.A., Seal, D.C.: Positivity-preserving discontinuous Galerkin methods with Lax-Wendroff time discretizations. J. Sci. Comput. 71, 44–70 (2017)
49. Nazarov, M.: Convergence of a residual based artificial viscosity finite element method. Comput. Math. Appl. 65(4), 616–626 (2013)
50. Pazner, W.: Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting. Comput. Methods Appl. Mech. Eng. 382, 113876 (2021)
51. Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2006-112 (2006)
52. Perthame, B., Shu, C.-W.: On positivity preserving finite volume schemes for Euler equations. Numer. Math. 73, 119–130 (1996)
53. Quezada de Luna, M., Ketcheson, D.I.: Maximum principle preserving space and time flux limiting for diagonally implicit Runge-Kutta discretizations of scalar convection-diffusion equations. J. Sci. Comput. 92, 102 (2022)
54. Ranocha, H.: Generalised Summation-by-Parts Operators and Entropy Stability of Numerical Methods for Hyperbolic Balance Laws. Cuvillier Verlag, Göttingen (2018)
55. Ranocha, H., Gassner, G.J.: Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes. Commun. Appl. Math. Comput. 4, 880–903 (2022)
56. Ranocha, H., Schlottke-Lakemper, M., Chan, J., Rueda-Ramírez, A.M., Winters, A.R., Hindenlang, F., Gassner, G.J.: Efficient implementation of modern entropy stable and kinetic energy preserving discontinuous Galerkin methods for conservation laws. arXiv:2112.10517 (2021)
57. Ranocha, H., Schlottke-Lakemper, M., Winters, A.R., Faulhaber, E., Chan, J., Gassner, G.: Adaptive numerical simulations with Trixi.jl: a case study of Julia for scientific computing. Proc. JuliaCon Confer. 1(1), 77 (2022)
58. Rueda-Ramírez, A.M., Gassner, G.J.: A subcell finite volume positivity-preserving limiter for DGSEM discretizations of the Euler equations. arXiv:2102.06017 (2021)
59. Rueda-Ramírez, A.M., Gassner, G.J.: A flux-differencing formula for split-form summation by parts discretizations of non-conservative systems: applications to subcell limiting for magneto-hydrodynamics. arXiv:2211.14009 (2022)
60. Rueda-Ramírez, A.M., Hennemann, S., Hindenlang, F.J., Winters, A.R., Gassner, G.J.: An entropy stable nodal discontinuous Galerkin method for the resistive mhd equations. Part II: subcell finite volume shock capturing. J. Comput. Phys. 444, 110580 (2021)
61. Rueda-Ramírez, A.M., Pazner, W., Gassner, G.J.: Subcell limiting strategies for discontinuous Galerkin spectral element methods. Comput. Fluids 247, 105627 (2022)
62. Schlottke-Lakemper, M., Gassner, G.J., Ranocha, H., Winters, A.R.: Trixi.jl: adaptive high-order numerical simulations of hyperbolic PDEs in Julia. https://github.com/trixi-framework/Trixi.jl (2020)
63. Schlottke-Lakemper, M., Winters, A.R., Ranocha, H., Gassner, G.J.: A purely hyperbolic discontinuous Galerkin approach for self-gravitating gas dynamics. J. Comput. Phys. 110467 (2021)
64. Sedov, L.I.: Similarity and Dimensional Methods in Mechanics. Academic Press, USA (1959)
65. Selmin, V.: Finite element solution of hyperbolic equations. II. Two-dimensional case. Research Report RR-0708, INRIA (1987)
66. Shima, N., Kuya, Y., Tamaki, Y., Kawai, S.: Preventing spurious pressure oscillations in split convective form discretization for compressible flows. J. Comput. Phys. 427, 110060 (2021)
67. Tadmor, E.: A minimum entropy principle in the gas dynamics equations. Appl. Numer. Math. 2(3/4/5), 211–219 (1986)
68. Tadmor, E.: Entropy functions for symmetric systems of conservation laws. J. Math. Anal. Appl. 122(2), 355–359 (1987)
69. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)
70. Thrussell, J., Lieberman, E., Ferguson, J.: Exactpack. Technical report, Los Alamos National Lab. (LANL), Los Alamos, NM (United States) (2023)
71. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction. Springer, Berlin (2013)
72. Vilar, F.: A posteriori correction of high-order discontinuous Galerkin scheme through subcell finite volume formulation and flux reconstruction. J. Comput. Phys. 387, 245–279 (2019)
73. Vilar, F., Abgrall, R.: A posteriori local subcell correction of high-order discontinuous Galerkin scheme for conservation laws on two-dimensional unstructured grids. arXiv:2212.11358 (2022)
74. Wu, X., Trask, N., Chan, J.: Entropy stable discontinuousGalerkinmethodsfor the shallow water equations with subcell positivity preservation. arXiv:2112.07749 (2021)
75. Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys.31(3), 335–362 (1979)
76. Zhang, X., Shu, C.-W.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229(23), 8918–8934 (2010)
77. Zhang, X., Xia, Y., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. J. Sci. Comput. 50(1), 29–62(2012)
78. Zhu, J., Qiu, J.: Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method, III: unstructured meshes. J. Sci. Comput. 39(2), 293–321 (2009)

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