Entropy-stable Gauss collocation methods for ideal magneto-hydrodynamics. (English) Zbl 07649272
Summary: In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms.
Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, J. Chan et al. [SIAM J. Sci. Comput. 41, No. 5, A2938–A2966 (2019; Zbl 1435.65172)] presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. [loc. cit.] to the non-conservative GLM-MHD system.
We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart.
Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, J. Chan et al. [SIAM J. Sci. Comput. 41, No. 5, A2938–A2966 (2019; Zbl 1435.65172)] presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. [loc. cit.] to the non-conservative GLM-MHD system.
We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76Mxx | Basic methods in fluid mechanics |
| 35Lxx | Hyperbolic equations and hyperbolic systems |
Keywords:
compressible magnetohydrodynamics; entropy stability; discontinuous Galerkin spectral element methods; Gauss nodesCitations:
Zbl 1435.65172References:
| [1] | 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 |
| [2] | Munz, C. D.; Omnes, P.; Schneider, R.; Sonnendrücker, E.; Voß, U., Divergence correction techniques for Maxwell solvers based on a hyperbolic model, J. Comput. Phys., 161, 484-511 (2000) · Zbl 0970.78010 |
| [3] | Dedner, A.; Kemm, F.; Kröner, D.; Munz, C. D.; Schnitzer, T.; Wesenberg, M., Hyperbolic divergence cleaning for the MHD equations, J. Comput. Phys., 175, 645-673 (2002) · Zbl 1059.76040 |
| [4] | Balsara, D. S.; Kumar, R.; Chandrashekar, P., Globally divergence-free dg scheme for ideal compressible mhd, Commun. Appl. Math. Comput. Sci., 16, 59-98 (2021) · Zbl 1468.65134 |
| [5] | Li, F.; Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for mhd equations, J. Sci. Comput., 22, 413-442 (2005) · Zbl 1123.76341 |
| [6] | Balsara, D. S., Second-order-accurate schemes for magnetohydrodynamics with divergence-free reconstruction, Astrophys. J. Suppl. Ser., 151, 149 (2004) |
| [7] | Fey, M.; Torrilhon, M., A constrained transport upwind scheme for divergence-free advection, (Hyperbolic Problems: Theory, Numerics, Applications (2003), Springer), 529-538 · Zbl 1134.76394 |
| [8] | Balsara, D. S.; Dumbser, M., Divergence-free mhd on unstructured meshes using high order finite volume schemes based on multidimensional Riemann solvers, J. Comput. Phys., 299, 687-715 (2015) · Zbl 1351.76092 |
| [9] | Balsara, D. S.; Rumpf, T.; Dumbser, M.; Munz, C.-D., Efficient, high accuracy ader-weno schemes for hydrodynamics and divergence-free magnetohydrodynamics, J. Comput. Phys., 228, 2480-2516 (2009) · Zbl 1275.76169 |
| [10] | Godunov, S. K., Symmetric form of the equations of magnetohydrodynamics, (Numerical Methods for Mechanics of Continuum Medium, vol. 1 (1972)), 26-34 |
| [11] | Powell, K. G., An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension) (1994), Technical Report |
| [12] | Powell, K. G.; Roe, P. L.; Linde, T. J.; Gombosi, T. I.; De Zeeuw, D. L., A solution-adaptive upwind scheme for ideal magnetohydrodynamics, J. Comput. Phys., 154, 284-309 (1999) · Zbl 0952.76045 |
| [13] | Derigs, D.; Winters, A. R.; Gassner, G. J.; Walch, S.; Bohm, M., Ideal GLM-MHD: about the entropy consistent nine-wave magnetic field divergence diminishing ideal magnetohydrodynamics equations, J. Comput. Phys., 364, 420-467 (2018) · Zbl 1392.76037 |
| [14] | Chandrashekar, P.; Klingenberg, C., Entropy stable finite volume scheme for ideal compressible MHD on 2-D Cartesian meshes, SIAM J. Numer. Anal., 54, 1313-1340 (2016) · Zbl 1381.76213 |
| [15] | Hindenlang, F.; Gassner, G. J.; Altmann, C.; Beck, A.; Staudenmaier, M.; Munz, C. D., Explicit discontinuous Galerkin methods for unsteady problems, Comput. Fluids, 61, 86-93 (2012) · Zbl 1365.76117 |
| [16] | Wang, Z. J.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A.; Deconinck, H.; Hartmann, R.; Hillewaert, K.; Huynh, H. T.; Kroll, N.; May, G.; Persson, P.-O.; van Leer, B.; Visbal, M.; van Leer, B.; Visbal, M., High-order CFD methods: current status and perspective, Int. J. Numer. Methods Fluids, 72, 811-845 (2013) · Zbl 1455.76007 |
| [17] | Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., The development of discontinuous Galerkin methods, (Discontinuous Galerkin Methods, vol. 11 (2000)), 3-50 · Zbl 0989.76045 |
| [18] | Rivière, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations Theory and Implementation (2008), SIAM · Zbl 1153.65112 |
| [19] | Kopriva, D. A.; Woodruff, S. L.; Hussaini, M. Y., Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, Int. J. Numer. Methods Eng., 53, 105-122 (2002) · Zbl 0994.78020 |
| [20] | Rueda-Ramírez, A. M.; Manzanero, J.; Ferrer, E.; Rubio, G.; Valero, E., A p-multigrid strategy with anisotropic p-adaptation based on truncation errors for high-order discontinuous Galerkin methods, J. Comput. Phys., 378, 209-233 (2019) · Zbl 1416.65357 |
| [21] | Rueda-Ramírez, A. M.; Rubio, G.; Ferrer, E.; Valero, E., Truncation error estimation in the p-anisotropic discontinuous Galerkin spectral element method, J. Sci. Comput., 78, 433-466 (2019) · Zbl 1410.65353 |
| [22] | 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, A1233-A1253 (2013) · Zbl 1275.65065 |
| [23] | Fisher, T. C.; Carpenter, M. H.; Nordström, J.; Yamaleev, N. K.; Swanson, C., Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: theory and boundary conditions, J. Comput. Phys., 234, 353-375 (2013) · Zbl 1284.65102 |
| [24] | 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, B835-B867 (2014) · Zbl 1457.65140 |
| [25] | Wintermeyer, N.; Winters, A. R.; Gassner, G. J.; Kopriva, D. A., An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry, J. Comput. Phys., 340, 200-242 (2017) · Zbl 1380.65291 |
| [26] | Manzanero, J.; Rubio, G.; Kopriva, D. A.; Ferrer, E.; Valero, E., An entropy-stable discontinuous Galerkin approximation for the incompressible Navier-Stokes equations with variable density and artificial compressibility, J. Comput. Phys., 408, Article 109241 pp. (2020) · Zbl 07505602 |
| [27] | 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 |
| [28] | 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 |
| [29] | Manzanero, J.; Rubio, G.; Kopriva, D. A.; Ferrer, E.; Valero, E., Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes/Cahn-Hilliard system, J. Comput. Phys., 408, Article 109363 pp. (2020) · Zbl 07505633 |
| [30] | Coquel, F.; Marmignon, C.; Rai, P.; Renac, F., An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model, J. Comput. Phys., 431, Article 110135 pp. (2021) · Zbl 07511452 |
| [31] | 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., 1, 1-35 (2018) · Zbl 07508372 |
| [32] | Rojas, D.; Boukharfane, R.; Dalcin, L.; Fernández, D. C.D. R.; Ranocha, H.; Keyes, D. E.; Parsani, M., On the robustness and performance of entropy stable discontinuous collocation methods, J. Comput. Phys., 426, Article 109891 pp. (2021) · Zbl 07510039 |
| [33] | Ranocha, H., Shallow water equations: split-form, entropy stable, well-balanced, and positivity preserving numerical methods, GEM Int. J. Geomath., 8, 85-133 (2017) · Zbl 1432.65156 |
| [34] | Abgrall, R.; Nordström, J.; Öffner, P.; Tokareva, S., Analysis of the SBP-SAT stabilization for finite element methods part I: linear problems, J. Sci. Comput., 85, 1-29 (2020) · Zbl 1456.65100 |
| [35] | 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 |
| [36] | 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 |
| [37] | 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 |
| [38] | Ortleb, S., Kinetic energy preserving dg schemes based on summation-by-parts operators on interior node distributions, PAMM, 16, 857-858 (2016) |
| [39] | Ortleb, S., A kinetic energy preserving dg scheme based on Gauss-Legendre points, J. Sci. Comput., 71, 1135-1168 (2017) · Zbl 1516.65095 |
| [40] | Del Rey Fernández, D. C.; Boom, P. D.; Carpenter, M. H.; Zingg, D. W., Extension of tensor-product generalized and dense-norm summation-by-parts operators to curvilinear coordinates, J. Sci. Comput., 80, 1957-1996 (2019) · Zbl 1428.65004 |
| [41] | Hicken, J. E.; Del Rey Fernández, D. C.; Zingg, D. W., Multidimensional summation-by-parts operators: general theory and application to simplex elements, SIAM J. Sci. Comput., 38, A1935-A1958 (2016) · Zbl 1382.65355 |
| [42] | Chan, J.; Ranocha, H.; Rueda-Ramírez, A. M.; Gassner, G.; Warburton, T., On the entropy projection and the robustness of high order entropy stable discontinuous Galerkin schemes for under-resolved flows, Front. Phys., 356 (2022) |
| [43] | Wu, K., Positivity-preserving analysis of numerical schemes for ideal magnetohydrodynamics, SIAM J. Numer. Anal., 56, 2124-2147 (2018) · Zbl 1391.76369 |
| [44] | Wu, K.; Shu, C.-W., Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes, Numer. Math., 142, 995-1047 (2019) · Zbl 1419.76446 |
| [45] | Rueda-Ramírez, A. M.; Ferrer, E.; Kopriva, D. A.; Rubio, G.; Valero, E., A statically condensed discontinuous Galerkin spectral element method on Gauss-Lobatto nodes for the compressible Navier-Stokes equations, J. Comput. Phys. (2020) · Zbl 07510067 |
| [46] | 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 (2021) · Zbl 07515469 |
| [47] | Ismail, F.; Roe, P. L., Affordable, entropy-consistent Euler flux functions II: entropy production at shocks, J. Comput. Phys., 228, 5410-5436 (2009) · Zbl 1280.76015 |
| [48] | Derigs, D.; Winters, A. R.; Gassner, G. J.; Walch, S., A novel averaging technique for discrete entropy-stable dissipation operators for ideal MHD, J. Comput. Phys., 330, 624-632 (2017) · Zbl 1378.76131 |
| [49] | 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 |
| [50] | 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) · Zbl 1349.65293 |
| [51] | 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., Article 109935 pp. (2020) · Zbl 07510051 |
| [52] | Tadmor, E., Entropy functions for symmetric systems of conservation laws, J. Math. Anal. Appl., 122, 355-359 (1987) · Zbl 0624.35057 |
| [53] | Tadmor, E., A minimum entropy principle in the gas dynamics equations, Appl. Numer. Math., 2, 211-219 (1986) · Zbl 0625.76084 |
| [54] | Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer., 12, 451-512 (2003) · Zbl 1046.65078 |
| [55] | Volpert, A., The spaces bv and quasilinear equations, Math. USSR Sb., 2, 225-267 (1967) · Zbl 0168.07402 |
| [56] | Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Numer. Anal., 44, 300-321 (2006) · Zbl 1130.65089 |
| [57] | Chan, J., On discretely entropy conservative and entropy stable discontinuous Galerkin methods, J. Comput. Phys., 362, 346-374 (2018) · Zbl 1391.76310 |
| [58] | Chan, J., Skew-symmetric entropy stable modal discontinuous Galerkin formulations, J. Sci. Comput., 81, 459-485 (2019) · Zbl 1466.65123 |
| [59] | Kopriva, D. A., Metric identities and the discontinuous spectral element method on curvilinear meshes, J. Sci. Comput., 26, 301-327 (2006) · Zbl 1178.76269 |
| [60] | Kopriva, D. A., Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers (2009), Springer Science & Business Media · Zbl 1172.65001 |
| [61] | Manzanero, J., A high-order discontinuous Galerkin multiphase flow solver for industrial applications (2020), Universidad Politécnica de Madrid, Ph.D. thesis |
| [62] | Carpenter, M. H.; Kennedy, C. A., Fourth-order 2N-storage Runge-Kutta schemes (1994), Technical Report |
| [63] | Hindenlang, F.; Bolemann, T.; Munz, C.-D., Mesh Curving Techniques for High Order Discontinuous Galerkin Simulations (2015), University of Stuttgart, Ph.D. thesis |
| [64] | 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 (2011) · Zbl 1255.65089 |
| [65] | Stone, J. M.; Gardiner, T. A.; Teuben, P.; Hawley, J. F.; Simon, J. B., Athena: a new code for astrophysical MHD, Astrophys. J. Suppl. Ser., 178, 137-177 (2008) |
| [66] | F. Hindenlang, G. Gassner, A new entropy conservative two-point flux for ideal mhd equations derived from first principles, Talk presented at HONOM, 2019. |
| [67] | Winters, A. R.; Gassner, G. J., Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations, J. Comput. Phys., 304, 72-108 (2016) · Zbl 1349.76407 |
| [68] | Mignone, A.; Tzeferacos, P.; Bodo, G., High-order conservative finite difference GLM-MHD schemes for cell-centered MHD, J. Comput. Phys., 229, 5896-5920 (2010) · Zbl 1425.76305 |
| [69] | T. Chen, C.-W. Shu, Review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes, 2020. |
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