Abstract
In this paper, we present three different numerical approaches to account for curl-type involution constraints in hyperbolic partial differential equations for continuum physics. All approaches have a direct analogy to existing and well-known divergence-preserving schemes for the Maxwell and MHD equations. The first method consists in a generalization of the Godunov–Powell terms, which means adding suitable multiples of the involution constraints to the PDE system in order to achieve the symmetric Godunov form. The second method is an extension of the generalized Lagrangian multiplier (GLM) approach of Munz et al., where the numerical errors in the involution constraint are propagated away via an augmented PDE system. The last method is an exactly involution-preserving discretization, similar to the exactly divergence-free schemes for the Maxwell and MHD equations, making use of appropriately staggered meshes. We present some numerical results that allow to compare all three approaches with each other.
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References
Alic, D., Bona, C., Bona-Casas, C.: Towards a gauge-polyvalent numerical relativity code. Phys. Rev. D 79(4), 044026 (2009)
Balsara, D.S.: Divergence-free adaptive mesh refinement for magnetohydrodynamics. J. Comput. Phys. 174(2), 614–648 (2001)
Balsara, D.S.: Second-order accurate schemes for magnetohydrodynamics with divergence-free reconstruction. Astrophys. J. Suppl. Ser. 151, 149–184 (2004)
Balsara, D.S.: Multidimensional HLLE Riemann solver: application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 229, 1970–1993 (2010)
Balsara, D.S.: Three dimensional HLL Riemann solver for conservation laws on structured meshes; application to Euler and magnetohydrodynamic flows. J. Comput. Phys. 295, 1–23 (2015)
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)
Balsara, D.S., Spicer, D.: A staggered mesh algorithm using high order Godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations. J. Comput. Phys. 149, 270–292 (1999)
Brown, J.D., Diener, P., Field, S.E., Hesthaven, J.S., Herrmann, F., Mroué, A.H., Sarbach, O., Schnetter, E., Tiglio, M., Wagman, M.: Numerical simulations with a first-order BSSN formulation of Einstein’s field equations. Phys. Rev. D 85(8), 084004 (2012)
Chiocchetti, S., Peshkov, I., Gavrilyuk, S., Dumbser, M.: High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension. J. Comput. Phys. (2020), submitted
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)
DeVore, C.R.: Flux-corrected transport techniques for multidimensional compressible magnetohydrodynamics. J. Comput. Phys. 92, 142–160 (1991)
Dhaouadi, F., Favrie, N., Gavrilyuk, S.: Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation. Stud. Appl. Math. 1–20 (2018)
Dumbser, M., Peshkov, I., Romenski, E., Zanotti, O.: High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. J. Comput. Phys. 314, 824–862 (2016)
Dumbser, M., Guercilena, F., Köppel, S., Rezzolla, L., Zanotti, O.: Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes. Phys. Rev. D 97, 084053 (2018)
Dumbser, M., Fambri, F., Gaburro, E., Reinarz, A.: On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations. J. Comput. Phys. 404, 109088 (2020). https://doi.org/10.1016/j.jcp.2019.109088
Gardiner, T.A., Stone, J.M.: An unsplit Godunov method for ideal MHD via constrained transport. J. Comput. Phys. 205, 509–539 (2005)
Godunov, S.K.: An interesting class of quasilinear systems. Dokl. Akad. Nauk SSSR 139(3), 521–523 (1961)
Godunov, S.K.: Symmetric form of the magnetohydrodynamic equation. Numer. Methods Mech. Contin. Medium 3(1), 26–34 (1972)
Godunov, S.K., Romenski, E.I.: Nonstationary equations of the nonlinear theory of elasticity in Euler coordinates. J. Appl. Mech. Tech. Phys. 13, 868–885 (1972)
Godunov, S.K., Romenski, E.I.: Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic/Plenum Publishers, New York (2003)
Hyman, J.M., Shashkov, M.: Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl. 33, 81–104 (1997)
Jeltsch, R., Torrilhon, M.: On curl-preserving finite volume discretizations for shallow water equations. BIT Numer. Math. 46, S35–S53 (2006)
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)
Peshkov, I., Romenski, E.: A hyperbolic model for viscous Newtonian flows. Contin. Mech. Thermodyn. 28, 85–104 (2016)
Peshkov, I., Pavelka, M., Romenski, E., Grmela, M.: Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations. Contin. Mech. Thermodyn. 30(6), 1343–1378 (2018)
Powell, K.G.: An approximate Riemann solver for magnetohydrodynamics (that works in more than one dimension). Technical Report ICASE-Report 94-24 (NASA CR-194902), NASA Langley Research Center, Hampton, VA (1994)
Romenski, E.I.: Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics. Math. Comput. Model. 28(10), 115–130 (1998)
Romenski, E., Resnyansky, A.D., Toro, E.F.: Conservative hyperbolic formulation for compressible two-phase flow with different phase pressures and temperatures. Q. Appl. Math. 65, 259–279 (2007)
Romenski, E., Drikakis, D., Toro, E.F.: Conservative models and numerical methods for compressible two-phase flow. J. Sci. Comput. 42, 68–95 (2010)
Schmidmayer, K., Petitpas, F., Daniel, E., Favrie, N., Gavrilyuk, S.: A model and numerical method for compressible flows with capillary effects. J. Comput. Phys. 334, 468–496 (2017)
Torrilhon, M., Fey, M.: Constraint-preserving upwind methods for multidimensional advection equations. SIAM J. Numer. Anal. 42, 1694–1728 (2004)
Yee, K.S.: Numerical solution of initial voundary value problems involving Maxwell equation in isotropic media. IEEE Trans. Antenna Propag. 14, 302–307 (1966)
Acknowledgements
The research presented in this paper has been funded by the European Union’s Horizon 2020 Research and Innovation Programme under the project ExaHyPE, grant no. 671698 and by the Deutsche Forschungsgemeinschaft (DFG) under the project DROPIT, grant no. GRK 2160/1. MD also acknowledges financial support from the Italian Ministry of Education, University and Research (MIUR) via the Departments of Excellence Initiative 2018–2022 attributed to DICAM of the University of Trento (grant L. 232/2016) and via the PRIN 2017 project.
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Dumbser, M., Chiocchetti, S., Peshkov, I. (2020). On Numerical Methods for Hyperbolic PDE with Curl Involutions. In: Demidenko, G., Romenski, E., Toro, E., Dumbser, M. (eds) Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov's Legacy. Springer, Cham. https://doi.org/10.1007/978-3-030-38870-6_17
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