Communications on Applied Mathematics and Computation >

2023 , Vol. 5 >Issue 2: 679 - 721

DOI: https://doi.org/10.1007/s42967-021-00120-x

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

Expand
  • 1 Department for Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany;
    2 Department for Mathematics and Computer Science, Center for Data and Simulation Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany;
    3 I. Physikalisches Institut, Center for Data and Simulation Science, University of Cologne, Zülpicher Str. 77, 50937 Cologne, Germany

Received date: 2019-12-18

  Revised date: 2021-01-08

  Online published: 2023-05-26

Supported by

JM acknowledgesfunding through theKlaus-Tschira Stiftung via the project "DG2RAV". GG thanks the Klaus-Tschira Stiftung and the European Research Council for funding through the ERC Starting Grant "An Exascale aware and Un-crashable Space-Time-Adaptive Discontinuous Spectral Element Solver for Non-Linear Conservation Laws" (EXTREME, project no. 71448). SW thanks the Klaus-Tschira Stiftung, and acknowledges the Deutsche Forschungsgemeinschaft (DFG) for funding the sub-project C5 in the SFB956 and the European Research Council for funding the ERC Starting Grant "The radiative interstellar medium" (RADFEEDBACK, project no. 679852). This work was performed on the Cologne High Efficiency Operating Platform for Sciences (CHEOPS) at the Regionales Rechenzentrum Köln (RRZK). Furthermore, we thank Dr. Andrés Rueda-Ramírez for his help with the generation of the 3D supernova plot in Fig. 17.

Fold

Abstract

In this paper, a new strategy for a sub-element-based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low-to-high-order discretizations on this set of data, including a first-order finite volume scheme up to the full-order DG scheme. The different DG discretizations are then blended according to sub-element troubled cell indicators, resulting in a final discretization that adaptively blends from low to high order within a single DG element. The goal is to retain as much high-order accuracy as possible, even in simulations with very strong shocks, as, e.g., presented in the Sedov test. The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing. The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

Key words： High-order methods; Discontinuous Galerkin spectral element method; Finite volume method; Shock capturing; Astrophysics; Stellar physics

Cite this article

Johannes Markert, Gregor Gassner, Stefanie Walch . A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2023 , 5(2) : 679 -721 . DOI: 10.1007/s42967-021-00120-x

References

1. Black, K.:A conservative spectral element method for the approximation of compressible fluid flow. Kybernetika 35(1), 133-146 (1999)
2. Bohm, M., Schermeng, S., Winters, A.R., Gassner, G.J., Jacobs, G.B.:Multi-element SIAC filter for shock capturing applied to high-order discontinuous Galerkin spectral element methods. J. Sci. Comput. 81(2), 820-844 (2019)
3. Bohm, M., Winters, A.R., Gassner, G.J., Derigs, D., Hindenlang, F., Saur, J.:An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I:theory and numerical verification. J. Comput. Phys. 422, 108076 (2020)
4. Burstedde, C., Wilcox, L.C., Ghattas, O.:p4est:Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM J. Sci. Comput. 33(3), 1103-1133 (2011)
5. Carpenter, M., Fisher, T., Nielsen, E., Frankel, S.:Entropy stable spectral collocation schemes for the Navier-Stokes equations:discontinuous interfaces. SIAM J. Sci. Comput. 36(5), B835-B867 (2014)
6. Chan, J.:On discretely entropy conservative and entropy stable discontinuous Galerkin methods. J. Comput. Phys. 362, 346-374 (2018)
7. Chen, T., Shu, C.-W.:Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427-461 (2017)
8. Chevalier, R.A.:Self-similar solutions for the interaction of stellar ejecta with an external medium. Astrophys. J. 258, 790-797 (1982)
9. Ching, E.J., Lv, Yu., Gnoffo, P., Barnhardt, M., Ihme, M.:Shock capturing for discontinuous Galerkin methods with application to predicting heat transfer in hypersonic flows. J. Comput. Phys. 376, 54-75 (2019)
10. Cockburn, B., Hou, S., Shu, C.-W.:The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV:the multidimensional case. Math. Comput. 54(190), 545-581 (1990)
11. Cockburn, B., Shu, C.-W.:The Runge-Kutta discontinuous Galerkin method for conservation laws V:multidimensional systems. J. Comput. Phys. 141(2), 199-224 (1998)
12. Dumbser, M., Fambri, F., Tavelli, M., Bader, M., Weinzierl, T.:Efficient implementation of ADER discontinuous Galerkin schemes for a scalable hyperbolic PDE engine. Axioms 7(3), 63 (2018)
13. Dumbser, M., Loubere, R.:A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes. J. Comput. Phys. 319, 163-199 (2016)
14. Dumbser, M., Zanotti, O., Loubere, 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)
15. Einfeldt, B.:On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal. 25(2), 294-318 (1988)
16. Ferrand, G., Decourchelle, A., Safi-Harb, S.:Three-dimensional simulations of the thermal X-ray emission from young supernova remnants including efficient particle acceleration. Astrophys. J. 760(1), 34 (2012)
17. Flad, D., Gassner, G.:On the use of kinetic energy preservaing DG-scheme for large eddy simulation. J. Comput. Phys. 350, 782-795 (2017)
18. Fraschetti, F., Teyssier, R., Ballet, J., Decourchelle, A.:Simulation of the growth of the 3d RayleighTaylor instability in supernova remnants using an expanding reference frame. Astron. Astrophys. 515, A104 (2010)
19. Friedrich, L., Winters, A.R., Del Rey, D.C., Fernández, G.J., Gassner, M.P., Carpenter, M.H.:An entropy stable h/p non-conforming discontinuous Galerkin method with the summation-by-parts property. J. Sci. Comput. 77(2), 689-725 (2018)
20. Fryxell, B., Olson, K., Ricker, P., Timmes, F.X., Zingale, M., Lamb, D.Q., MacNeice, P., Rosner, R., Truran, J.W., Tufo, H.:Flash:an adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes. Astrophys. J. Suppl. Ser. 131(1), 273 (2000)
21. Gassner, G.:A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), A1233-A1253 (2013)
22. Gassner, G.J.:A kinetic energy preserving nodal discontinuous Galerkin spectral element method. Int. J. Numer. Methods Fluids 76(1), 28-50 (2014)
23. Gassner, G.J., Beck, A.D.:On the accuracy of high-order discretizations for underresolved turbulence simulations. Theor. Comput. Fluid Dyn. 27(3/4), 221-237 (2013)
24. Gassner, G., Staudenmaier, M., Hindenlang, F., Atak, M., Munz, C.-D.:A space-time adaptive discontinuous Galerkin scheme. Comput. Fluids 117, 247-261 (2015)
25. Gassner, G.J., Winters, A.R., Hindenlang, F.J., Kopriva, D.A.:The BR1 scheme is stable for the compressible Navier-Stokes equations. J. Sci. Comput. 77, 154-200 (2018)
26. 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)
27. Guo, W., Nair, R.D., Zhong, X.:An efficient WENO limiter for discontinuous Galerkin transport scheme on the cubed sphere. Int. J. Numer. Methods Fluids 81(1), 3-21 (2015)
28. 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)
29. Hesthaven, J.S., Warburton, T.:Nodal Discontinuous Galerkin Methods. Springer, Berlin (2008)
30. Kenevsky, A.:High-order implicit-explicit Runge-Kutta time integration schemes and time-consistent filtering in spectral methods. Ph.D. dissertation, Brown University, USA (2006)
31. Kirby, R.M., Karniadakis, G.E.:De-aliasing on non-uniform grids:algorithms and applications. J. Comput. Phys. 191, 249-264 (2003)
32. Klöckner, A., Warburton, T., Hesthaven, J.S.:Viscous shock capturing in a time-explicit discontinuous Galerkin method. Math. Model. Nat. Phenom. 6(3), 57-83 (2011)
33. Kolda, T.G., Bader, B.W.:Tensor decompositions and applications. SIAM Rev. 51(3), 455-500 (2009)
34. Kopriva, D.A.:Implementing Spectral Methods for Partial Differential Equations:Algorithms for Scientists and Engineers. Springer Science & Business Media, Berlin (2009)
35. Kopriva, D.A.:Stability of overintegration methods for nodal discontinuous Galerkin spectral element methods. J. Sci. Comput. 76(1), 426-442 (2017)
36. Kopriva, D.A., Hindenlang, F.J., Bolemann, T., Gassner, G.J.:Free-stream preservation for curved geometrically non-conforming discontinuous Galerkin spectral elements. J. Sci. Comput. 79(3), 1389-1408 (2019)
37. Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.:Computation of electromagnetic scattering with a nonconforming discontinuous spectral element method. Int. J. Numer. Methods Eng. 53(1), 105-122 (2002)
38. Korobeinikov, V.P.:Problems of Point Blast Theory. Springer Science & Business Media, Berlin (1991)
39. Krais, N., Beck, A., Bolemann, T., Frank, H., Flad, D., Gassner, G., Hindenlang, F., Hoffmann, M., Kuhn, T., Sonntag, M., Munz, C.-D.:FLEXI:a high order discontinuous Galerkin framework for hyperbolicparabolic conservation laws. Comput. Math. Appl. (2020)
40. Kuzmin, D.:Slope limiting for discontinuous Galerkin approximations with a possibly non-orthogonal Taylor basis. Int. J. Numer. Methods Fluids 71(9), 1178-1190 (2012)
41. Lax, P.D., Liu, X.-D.:Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM J. Sci. Comput. 19(2), 319-340 (1998)
42. Liu, Y., Shu, C.-W., Zhang, M.:Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes. J. Comput. Phys. 354, 163-178 (2018)
43. Murman, S.M., Diosady, L., Garai, A., Ceze, M.:A space-time discontinuous-Galerkin approach for separated flows. In:54th AIAA Aerospace Sciences Meeting, AIAA 2016-1059, AIAA Inc., Reston, VA (2016)
44. Parsani, M., Carpenter, M.H., Fisher, T.C., Nielsen, E.J.:Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier-Stokes equations. SIAM J. Sci. Comput. 38(5), A3129-A3162 (2016)
45. Parsani, M., Carpenter, M.H., Nielsen, E.J.:Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations. J. Comput. Phys. 290(C), 132-138 (2015)
46. Pazner, W., Persson, P.-O.:Analysis and entropy stability of the line-based discontinuous Galerkin method. J. Sci. Comput. 80(1), 376-402 (2019)
47. Persson, P.-O., Peraire, J.:Sub-cell shock capturing for discontinuous Galerkin methods. In:44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2006-112, AIAA Inc., Reston, VA (2006)
48. Peter, D.L.:Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun. Pure Appl. Math. 7(1), 159-193 (1954)
49. Phillip, C., Paul, R.W.:The piecewise parabolic method (ppm) for gas-dynamical simulations. J. Comput. Phys. 54(1), 174-201 (1984)
50. Qiu, J., Zhu, J.:RKDG with WENO type limiters. In:Kroll, N., et al. (eds) ADIGMA-A European Initiative on the Development of Adaptive Higher-Order Variational Methods for Aerospace Applications. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 113, pp. 67-80. Springer, Berlin (2010)
51. Roy, C.J., Nelson, C.C., Smith, T.M., Ober, C.C.:Verification of Euler/Navier-Stokes codes using the method of manufactured solutions. Int. J. Numer. Methods Fluids 44(6), 599-620 (2004)
52. Schulz-Rinne, C.W.:Classification of the Riemann problem for two-dimensional gas dynamics. SIAM J. Math. Anal. 24(1), 76-88 (1993)
53. Sedov, L.I.:Similarity and Dimensional Methods in Mechanics. Translation from 4th Russian edition, Academic Press, New York (1959)
54. Shu, C.-W., Osher, S.:Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439-471 (1988)
55. Sod, G.A.:A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27(1), 1-31 (1978)
56. Sonntag, M., Munz, C.-D.:Shock capturing for discontinuous Galerkin methods using finite volume subcells. In:Fuhrmann, J., Ohlberger, M., Rohde, C. (eds) Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems. Springer Proceedings in Mathematics & Statistics, vol. 78, pp. 945-953. Springer, Cham (2014)
57. Spiegel, S.C., Huynh, H.T., DeBonis, J.R.:De-aliasing through over-integration applied to the flux reconstruction and discontinuous Galerkin methods. In:22nd AIAA Computational Fluid Dynamics Conference, AIAA 2015-2744, AIAA Inc., Reston, VA (2015)
58. Spiteri, R.J., Ruuth, S.J.:A new class of optimal high-order strong-stability-preserving time discretization methods. SIAM J. Numer. Anal. 40(2), 469-491 (2002)
59. 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)
60. Wintermeyer, N., Winters, A.R., Gassner, G.J., Warburton, T.:An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. J. Comput. Phys. 375, 447-480 (2018)
61. Winters, A.R., Moura, R.C., Mengaldo, G., Gassner, G.J., Walch, S., Peiro, J., Sherwin, S.J.:A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. J. Comput. Phys. 372, 1-21 (2018)
62. Zanotti, O., Fambri, F., Dumbser, M., Hidalgo, A.:Space-time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting. Comput. Fluids 118, 204-224 (2015)
63. 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)
64. Zhang, X., Shu, C.-W.:Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231(5), 2245-2258 (2012)
65. 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)
66. Zingan, V., Guermond, J.-L., Morel, J., Popov, B.:Implementation of the entropy viscosity method with the discontinuous Galerkin method. Comput. Methods Appl. Mech. Eng. 253, 479-490 (2013)
Options
Outlines

/

〈 〉

Copyright © Editorial By Communications on Applied Mathematics and Computation
沪ICP备09014157