×

A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations. (English) Zbl 07510051

Summary: The main result in this paper is a provably entropy stable shock capturing approach for the high order entropy stable Discontinuous Galerkin Spectral Element Method (DGSEM) based on a hybrid blending with a subcell low order variant. Since it is possible to rewrite a high order summation-by-parts (SBP) operator into an equivalent conservative finite volume form, we were able to design a low order scheme directly with the Legendre-Gauss-Lobatto (LGL) nodes that is compatible to the discrete entropy analysis used for the proof of the entropy stable DGSEM. Furthermore, we present a hybrid low order/high order discretisation where it is possible to seamlessly blend between the two approaches, while still being provably entropy stable. With tensor products and careful design of the low order scheme on curved elements, we are able to extend the approach to three spatial dimensions on unstructured curvilinear hexahedral meshes. We validate our theoretical findings and demonstrate convergence order for smooth problems, conservation of the primary quantities and discrete entropy stability for an arbitrary blending on curvilinear grids. In practical simulations, we connect the blending factor to a local troubled element indicator that provides the control of the amount of low order dissipation injected into the high order scheme. We modified an existing shock indicator, which is based on the modal polynomial representation, to our provably stable hybrid scheme. The aim is to reduce the impact of the parameters as good as possible. We describe our indicator in detail and demonstrate its robustness in combination with the hybrid scheme, as it is possible to compute all the different test cases without changing the indicator. The test cases include e.g. the double Mach reflection setup, forward and backward facing steps with shock Mach numbers up to 100. The proposed approach is relatively straight forward to implement in an existing entropy stable DGSEM code as only modifications local to an element are necessary.

MSC:

65-XX Numerical analysis
76-XX Fluid mechanics

Software:

FLASH; HOPR
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Kirby, R. M.; Karniadakis, G., De-aliasing on non-uniform grids: algorithms and applications, J. Comput. Phys., 191, 249-264 (2003) · Zbl 1161.76534
[2] Mengaldo, G.; Grazia, D. D.; Moxey, D.; Vincent, P. E.; Sherwin, S. J., Dealiasing techniques for high-order spectral element methods on regular and irregular grids, J. Comput. Phys., 299, 56-81 (2015) · Zbl 1352.65396
[3] Gassner, G. J.; Beck, A. D., On the accuracy of high-order discretizations for underresolved turbulence simulations, Theor. Comput. Fluid Dyn., 27, 221-237 (2013)
[4] Beck, A. D.; Bolemann, T.; Flad, D.; Frank, H.; Gassner, G. J.; Hindenlang, F.; Munz, C.-D., High-order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations, Int. J. Numer. Methods Fluids, 76, 522-548 (2014)
[5] Kopriva, D. A., Stability of overintegration methods for nodal discontinuous Galerkin spectral element methods, J. Sci. Comput., 76, 426-442 (2017) · Zbl 1404.65174
[6] 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, B835-B867 (2014) · Zbl 1457.65140
[7] 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, A3129-A3162 (2016) · Zbl 1457.65149
[8] Chan, J., On discretely entropy conservative and entropy stable discontinuous Galerkin methods, J. Comput. Phys., 362, 346-374 (2018) · Zbl 1391.76310
[9] Parsani, M.; Carpenter, M.; Nielsen, E., Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations, J. Comput. Phys., 290, 132-138 (2015) · Zbl 1349.76250
[10] 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) · Zbl 1422.65280
[11] 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) · Zbl 1407.65189
[12] Hiltebrand, A.; Mishra, S., Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numer. Math., 126, 103-151 (2013) · Zbl 1303.65083
[13] Murman, S. M.; Diosady, L.; Garai, A.; Ceze, M., A space-time discontinuous-Galerkin approach for separated flows, (54th AIAA Aerospace Sciences Meeting (2016)), 1059
[14] 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) · Zbl 1415.76461
[15] 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) · Zbl 1416.65363
[16] Bohm, M., An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations (2018), Universität zu Köln, Ph.D. thesis
[17] Persson, P.-O.; Peraire, J., Sub-cell shock capturing for discontinuous Galerkin methods, 2006, pp. 1-13
[18] Lodato, G., Characteristic modal shock detection for discontinuous finite element methods, Comput. Fluids, 179, 309-333 (2019) · Zbl 1411.76067
[19] Klöckner, A.; Warburton, T.; Hesthaven, J., Viscous shock capturing in a time-explicit discontinuous Galerkin method, Math. Model. Nat. Phenom., 6 (2011) · Zbl 1220.65165
[20] Sonntag, M.; Munz, C.-D., Efficient parallelization of a shock capturing for discontinuous Galerkin methods using finite volume sub-cells, J. Sci. Comput., 70, 1262-1289 (2017) · Zbl 1366.65089
[21] 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) · Zbl 1452.65251
[22] Fisher, T.; Carpenter, M., High-order entropy stable finite difference schemes for nonlinear conservation laws. Finite domains (2013), NASA Technical Report · Zbl 1349.65293
[23] Gassner, G.; Kopriva, D. A., A comparison of the dispersion and dissipation errors of Gauss and Gauss-Lobatto discontinuous Galerkin spectral element methods, SIAM J. Sci. Comput., 33, 2560-2579 (2010) · Zbl 1255.65089
[24] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (1999), Springer Verlag · Zbl 0923.76004
[25] Gassner, G., A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput., 35, A1233-A1253 (2013) · Zbl 1275.65065
[26] Strand, B., Summation by parts for finite difference approximations for \(d / d x\), J. Comput. Phys., 110 (1994) · Zbl 0792.65011
[27] Harten, A., On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., 49, 151-164 (1982) · Zbl 0503.76088
[28] Ray, D.; Chandrashekar, P., Entropy stable schemes for compressible Euler equations, Int. J. Numer. Anal. Model. Ser. B, 4, 335-352 (2013) · Zbl 1463.76036
[29] Fisher, T., High-order \(L^2\) stable multi-domain finite difference method for compressible flows (2012), Purdue University, Ph.D. thesis
[30] Carpenter, M.; Fisher, T.; Nielsen, E.; Parsani, M.; Svärd, M.; Yamaleev, N., Chapter 19 - entropy stable summation-by-parts formulations for compressible computational fluid dynamics, (Abgrall, R.; Shu, C.-W., Handbook of Numerical Methods for Hyperbolic Problems. Handbook of Numerical Methods for Hyperbolic Problems, Handbook of Numerical Analysis, vol. 17 (2016), Elsevier), 495-524 · Zbl 1352.65001
[31] Chandrashekar, P., Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations (2012), CoRR
[32] Carpenter, M.; Kennedy, C., Fourth-order 2n-storage Runge-Kutta schemes (1994), Nasa reports TM, 109112
[33] (2020), DLR - Institute of Propulsion Technology - Department of Numerical Methods, TRACE user guide
[34] (2020), FLUXO
[35] F. Hindenlang, G. Gassner, D. Kopriva, Stability of wall boundary condition procedures for discontinuous Galerkin spectral element approximations of the compressible Euler equations, 2019.
[36] Cenaero, 5th international workshop on high-order CFD methods (2017)
[37] Chan, J.; Fernandez, D. C.D. R.; Carpenter, M. H.; Del Rey Fernández, D. C.; Carpenter, M. H., Efficient entropy stable Gauss collocation methods, SIAM J. Sci. Comput., 41, A2938-A2966 (2019) · Zbl 1435.65172
[38] Hindenlang, F.; Bolemann, T.; Munz, C.-D., Mesh curving techniques for high order discontinuous Galerkin simulations, (IDIHOM: Industrialization of High-Order Methods-a Top-Down Approach (2015), Springer), 133-152
[39] Ames Resarch Staff, National Advisory Committee for Aeronautics (1951), Report 1135 - equations, tables and charts for compressible flow
[40] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, J. Comput. Phys., 83, 32-78 (1989) · Zbl 0674.65061
[41] Colella, P.; Woodward, P. R., The piecewise parabolic method (ppm) for gas-dynamical simulations, J. Comput. Phys., 54, 174-201 (1984) · Zbl 0531.76082
[42] Fryxell, B.; Olson, K.; Ricker, P.; Timmes, F.; Zingale, M.; Lamb, D.; MacNeice, P.; Rosner, R.; Truran, J.; Tufo Flash, H., An adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes, Astrophys. J. Suppl. Ser., 131, 273 (2000)
[43] Takayama, K.; Inoue, O., Shock wave diffraction over a 90 degree sharp corner – posters presented at 18th ISSW, Shock Waves, 1, 301-312 (1991)
[44] Hillier, R., Computation of shock wave diffraction at a ninety degrees convex edge, Shock Waves, 1, 89-98 (1991) · Zbl 0825.76402
[45] Bagabir, A., Comparison of compression and blast waves diffraction over 90° sharp corner, Aljouf Univ. Sci. Eng. J., 3 (2016)
[46] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 5, 115-173 (1984) · Zbl 0573.76057
[47] Kemm, F., On the proper setup of the double Mach reflection as a test case for the resolution of gas dynamics codes, Comput. Fluids, 132, 72-75 (2016) · Zbl 1390.76275
[48] Fjordholm, U. S., High-order accurate entropy stable numerical schemes for hyperbolic conservation laws (2013), ETH Zurich: ETH Zurich Zürich, Ph.D. thesis
[49] Chen, T.; Shu, C.-W., Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, J. Comput. Phys., 345 (2017) · Zbl 1380.65253
[50] Kopriva, D. A., Metric identities and the discontinuous spectral element method on curvilinear meshes, J. Sci. Comput., 26, 301 (2006) · Zbl 1178.76269
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.