A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations. (English) Zbl 07649271
Summary: High-order entropy-stable discontinuous Galerkin methods for the compressible Euler and Navier-Stokes equations require the positivity of thermodynamic quantities in order to guarantee their well-posedness. In this work, we introduce a positivity limiting strategy for entropy-stable discontinuous Galerkin discretizations constructed by blending high order solutions with a low order positivity-preserving discretization. The proposed low order discretization is semi-discretely entropy stable, and the proposed limiting strategy is positivity preserving for the compressible Euler and Navier-Stokes equations. Numerical experiments confirm the high order accuracy and robustness of the proposed strategy.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 35Lxx | Hyperbolic equations and hyperbolic systems |
| 76Mxx | Basic methods in fluid mechanics |
Keywords:
high order; discontinuous Galerkin; entropy stable; compressible Euler; compressible Navier-Stokes; positivity preservingReferences:
| [1] | Slotnick, J.; Khodadoust, A.; Alonso, J.; Darmofal, D.; Gropp, W.; Lurie, E.; Mavriplis, D., CFD vision 2030 study: a path to revolutionary computational aerosciences (2013) |
| [2] | Wang, Z. J.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A.; Deconinck, H.; Hartmann, R.; Hillewaert, K.; Huynh, H. T., High-order CFD methods: current status and perspective, Int. J. Numer. Methods Fluids, 72, 8, 811-845 (2013) · Zbl 1455.76007 |
| [3] | Hesthaven, J. S.; Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (2007), Springer Science & Business Media |
| [4] | Chan, J., On discretely entropy conservative and entropy stable discontinuous Galerkin methods, J. Comput. Phys., 362, 346-374 (2018) · Zbl 1391.76310 |
| [5] | Chan, J., Skew-symmetric entropy stable modal discontinuous Galerkin formulations, J. Sci. Comput., 81, 1, 459-485 (2019) · Zbl 1466.65123 |
| [6] | Chan, J.; Lin, Y.; Warburton, T., Entropy stable modal discontinuous Galerkin schemes and wall boundary conditions for the compressible Navier-Stokes equations, J. Comput. Phys., 448, Article 110723 pp. (2022) · Zbl 1537.65127 |
| [7] | Carpenter, M. H.; Fisher, T. C.; Nielsen, E. J.; Frankel, S. H., Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces, SIAM J. Sci. Comput., 36, 5, B835-B867 (2014) · Zbl 1457.65140 |
| [8] | 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, A1233-A1253 (2013) · Zbl 1275.65065 |
| [9] | 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 |
| [10] | 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) · Zbl 1380.65253 |
| [11] | Löhner, R., Applied Computational Fluid Dynamics Techniques: an Introduction Based on Finite Element Methods (2008), John Wiley & Sons · Zbl 1151.76002 |
| [12] | Berthon, C., An invariant domain preserving MUSCL scheme, (Progress in Industrial Mathematics at ECMI 2006 (2008), Springer), 933-938 · Zbl 1308.65138 |
| [13] | 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) · Zbl 1282.76161 |
| [14] | Calgaro, C.; Creusé, E.; Goudon, T.; Penel, Y., Positivity-preserving schemes for Euler equations: sharp and practical CFL conditions, J. Comput. Phys., 234, 417-438 (2013) · Zbl 1284.65110 |
| [15] | Kuzmin, D.; Löhner, R.; Turek, S., Flux-Corrected Transport: Principles, Algorithms, and Applications (2012), Springer |
| [16] | Lohmann, C.; Kuzmin, D., Synchronized flux limiting for gas dynamics variables, J. Comput. Phys., 326, 973-990 (2016) · Zbl 1373.76286 |
| [17] | 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) · Zbl 1380.65279 |
| [18] | Kuzmin, D.; de Luna, M. Q., Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws, J. Comput. Phys., 411, Article 109411 pp. (2020) · Zbl 1436.65141 |
| [19] | 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) · Zbl 1346.65050 |
| [20] | 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, A3211-A3239 (2018) · Zbl 1402.65110 |
| [21] | 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) · Zbl 1440.65136 |
| [22] | Guermond, J.-L.; Maier, M.; Popov, B.; Tomas, I., Second-order invariant domain preserving approximation of the compressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 375, Article 113608 pp. (2021) · Zbl 1506.76076 |
| [23] | Zhang, X.; Shu, C.-W., On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229, 9, 3091-3120 (2010) · Zbl 1187.65096 |
| [24] | Zhang, X.; Shu, C.-W., Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments, Proc. R. Soc. A, Math. Phys. Eng. Sci., 467, 2134, 2752-2776 (2011) · Zbl 1222.65107 |
| [25] | 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) · Zbl 1282.76128 |
| [26] | Zhang, X., On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations, J. Comput. Phys., 328, 301-343 (2017) · Zbl 1406.65091 |
| [27] | Srinivasan, S.; Poggie, J.; Zhang, X., A positivity-preserving high order discontinuous Galerkin scheme for convection-diffusion equations, J. Comput. Phys., 366, 120-143 (2018) · Zbl 1406.65090 |
| [28] | Upperman, J.; Yamaleev, N. K., First-order positivity-preserving entropy stable spectral collocation scheme for the 3-D compressible Navier-Stokes equations (2021), preprint |
| [29] | 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) · Zbl 1349.65448 |
| [30] | Sonntag, M.; Munz, C.-D., Shock capturing for discontinuous Galerkin methods using finite volume subcells, (Finite Volumes for Complex Applications VII-Elliptic, Parabolic and Hyperbolic Problems (2014), Springer), 945-953 · Zbl 1426.76429 |
| [31] | 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 |
| [32] | 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, Article 109935 pp. (2021) · Zbl 07510051 |
| [33] | Rueda-Ramírez, A. M.; Gassner, G. J., A subcell finite volume positivity-preserving limiter for DGSEM discretizations of the Euler equations (2021), preprint |
| [34] | Pazner, W., Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting, Comput. Methods Appl. Mech. Eng., 382, Article 113876 pp. (2021) · Zbl 1506.65159 |
| [35] | Dzanic, T.; Witherden, F., Positivity-preserving entropy-based adaptive filtering for discontinuous spectral element methods, J. Comput. Phys., 468, Article 111501 pp. (2022) · Zbl 07578908 |
| [36] | Hajduk, H., Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws, Comput. Math. Appl., 87, 120-138 (2021) · Zbl 1524.65539 |
| [37] | Chan, J.; Demkowicz, L.; Moser, R., A DPG method for steady viscous compressible flow, Comput. Fluids, 98, 69-90 (2014) · Zbl 1391.76311 |
| [38] | Nishida, T., Global solution for an initial boundary value problem of a quasilinear hyperbolic system, Proc. Jpn. Acad., 44, 7, 642-646 (1968) · Zbl 0167.10301 |
| [39] | Chueh, K. N.; Conley, C. C.; Smoller, J. A., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26, 2, 373-392 (1977) · Zbl 0368.35040 |
| [40] | Lions, P.-L.; Perthame, B.; Souganidis, P. E., Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Commun. Pure Appl. Math., 49, 6, 599-638 (1996) · Zbl 0853.76077 |
| [41] | Guermond, J.-L.; Popov, B., Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math., 74, 2, 284-305 (2014) · Zbl 1446.76147 |
| [42] | Svärd, M., A convergent numerical scheme for the compressible Navier-Stokes equations, SIAM J. Numer. Anal., 54, 3, 1484-1506 (2016) · Zbl 1381.76243 |
| [43] | Hughes, T. J.; Franca, L.; Mallet, M., A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comput. Methods Appl. Mech. Eng., 54, 2, 223-234 (1986) · Zbl 0572.76068 |
| [44] | Chan, J.; Wilcox, L. C., On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes, J. Comput. Phys., 378, 366-393 (2019) · Zbl 1416.65340 |
| [45] | Crean, J.; Hicken, J. E.; Fernández, D. C.D. R.; Zingg, D. W.; Carpenter, M. H., High-order, entropy-stable discretizations of the Euler equations for complex geometries, (23rd AIAA Computational Fluid Dynamics Conference (2017), American Institute of Aeronautics and Astronautics) |
| [46] | Svärd, M.; Nordström, J., Review of summation-by-parts schemes for initial-boundary-value problems, J. Comput. Phys., 268, 17-38 (2014) · Zbl 1349.65336 |
| [47] | Fernández, D. C.D. R.; Hicken, J. E.; Zingg, D. W., Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. Fluids, 95, 171-196 (2014) · Zbl 1390.65064 |
| [48] | Wu, X.; Kubatko, E. J.; Chan, J., High-order entropy stable discontinuous Galerkin methods for the shallow water equations: curved triangular meshes and GPU acceleration, Comput. Math. Appl., 82, 179-199 (2021) · Zbl 1524.65613 |
| [49] | Wu, X.; Trask, N.; Chan, J., Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation (2021) |
| [50] | Trask, N.; Bochev, P.; Perego, M., A conservative, consistent, and scalable meshfree mimetic method, J. Comput. Phys., 409, Article 109187 pp. (2020) · Zbl 1435.65221 |
| [51] | Tadmor, E., The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. Comput., 49, 179, 91-103 (1987) · Zbl 0641.65068 |
| [52] | Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer., 12, 1, 451-512 (2003) · Zbl 1046.65078 |
| [53] | 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) · Zbl 1373.76121 |
| [54] | 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, 1, 154-200 (2018) · Zbl 1407.65189 |
| [55] | Harten, A.; Lax, P. D.; Leer, B.v., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61 (1983) · Zbl 0565.65051 |
| [56] | Boris, J. P.; Book, D. L., Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11, 1, 38-69 (1973) · Zbl 0251.76004 |
| [57] | Friedrich, L.; Schnücke, G.; Winters, A. R.; Fernández, D. C.; Gassner, G. J.; Carpenter, M. H., Entropy stable space-time discontinuous Galerkin schemes with summation-by-parts property for hyperbolic conservation laws, J. Sci. Comput., 80, 1, 175-222 (2019) · Zbl 1421.35262 |
| [58] | Ranocha, H.; Sayyari, M.; Dalcin, L.; Parsani, M.; Ketcheson, D. I., Relaxation Runge-Kutta methods: fully discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations, SIAM J. Sci. Comput., 42, 2, A612-A638 (2020) · Zbl 1432.76207 |
| [59] | Persson, P.-O.; Peraire, J., Sub-cell shock capturing for discontinuous Galerkin methods, (44th AIAA Aerospace Sciences Meeting and Exhibit (2006)), 112 |
| [60] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 2, 439-471 (1988) · Zbl 0653.65072 |
| [61] | Ismail, F.; Roe, P. L., Affordable, entropy-consistent Euler flux functions II: entropy production at shocks, J. Comput. Phys., 228, 15, 5410-5436 (2009) · Zbl 1280.76015 |
| [62] | Davis, S., Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comput., 9, 3, 445-473 (1988) · Zbl 0645.65050 |
| [63] | 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) · Zbl 1349.76769 |
| [64] | van der Vegt, J. J.; Ven, H., Slip flow boundary conditions in discontinuous Galerkin discretizations of the Euler equations of gas dynamics (2002) |
| [65] | Hindenlang, F. J.; Gassner, G. J.; Kopriva, D. A., Stability of wall boundary condition procedures for discontinuous Galerkin spectral element approximations of the compressible Euler equations, Lect. Notes Comput. Sci. Eng., 134, 3-19 (2020) · Zbl 1484.65256 |
| [66] | Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, II, (Upwind and High-Resolution Schemes (1989), Springer), 328-374 |
| [67] | Shu, C.-W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51, 1, 82-126 (2009) · Zbl 1160.65330 |
| [68] | 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) |
| [69] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 1, 115-173 (1984) · Zbl 0573.76057 |
| [70] | Daru, V.; Tenaud, C., Evaluation of TVD high resolution schemes for unsteady viscous shocked flows, Comput. Fluids, 30, 1, 89-113 (2000) · Zbl 0994.76070 |
| [71] | Daru, V.; Tenaud, C., Numerical simulation of the viscous shock tube problem by using a high resolution monotonicity-preserving scheme, Comput. Fluids, 38, 3, 664-676 (2009) · Zbl 1193.76092 |
| [72] | Guermond, J.-L.; Kronbichler, M.; Maier, M.; Popov, B.; Tomas, I., On the implementation of a robust and efficient finite element-based parallel solver for the compressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 389, Article 114250 pp. (2022) · Zbl 1507.76105 |
| [73] | Chan, J.; Del Rey Fernandez, D. C.; Carpenter, M. H., Efficient entropy stable Gauss collocation methods, SIAM J. Sci. Comput., 41, 5, A2938-A2966 (2019) · Zbl 1435.65172 |
| [74] | Towns, J.; Cockerill, T.; Dahan, M.; Foster, I.; Gaither, K.; Grimshaw, A.; Hazlewood, V.; Lathrop, S.; Lifka, D.; Peterson, G. D.; Roskies, R.; Scott, J.; Wilkins-Diehr, N., XSEDE: accelerating scientific discovery, Comput. Sci. Eng., 16, 05, 62-74 (2014) |
| [75] | Crean, J.; Hicken, J. E.; Fernández, D. C.D. R.; Zingg, D. W.; Carpenter, M. H., Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements, J. Comput. Phys., 356, 410-438 (2018) · Zbl 1380.76080 |
| [76] | Harten, A.; Hyman, J. M.; Lax, P. D.; Keyfitz, B., On finite-difference approximations and entropy conditions for shocks, Commun. Pure Appl. Math., 29, 3, 297-322 (1976) · Zbl 0351.76070 |
| [77] | 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, Article 114428 pp. (2022) · Zbl 1507.65186 |
| [78] | Renac, F., Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows, J. Comput. Phys., 382, 1-26 (2019) · Zbl 1451.76073 |
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.
