A well balanced finite volume scheme for general relativity. (English) Zbl 07456293


65-XX Numerical analysis
35L40 First-order hyperbolic systems
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C10 Equations of motion in general relativity and gravitational theory
85-08 Computational methods for problems pertaining to astronomy and astrophysics
85-10 Mathematical modeling or simulation for problems pertaining to astronomy and astrophysics
85A30 Hydrodynamic and hydromagnetic problems in astronomy and astrophysics


WhiskyMHD; Cosmos++
Full Text: DOI arXiv


[1] M. Alcubierre, Introduction to 3+1 Numerical Relativity, Internat. Ser. Mongr. Phys. 140, Oxford University Press, 2008. · Zbl 1140.83002
[2] D. Alic, C. Bona, and C. Bona-Casas, Towards a gauge-polyvalent numerical relativity code, Phys. Rev. D, 79 (2009), 044026.
[3] D. Alic, C. Bona-Casas, C. Bona, L. Rezzolla, and C. Palenzuela, Conformal and covariant formulation of the Z4 system with constraint-violation damping, Phys. Rev. D, 85 (2012), 064040.
[4] D. Alic, W. Kastaun, and L. Rezzolla, Constraint damping of the conformal and covariant formulation of the Z4 system in simulations of binary neutron stars, Phys. Rev. D, 88 (2013), 064049.
[5] M.-Á. Aloy and I. Cordero-Carrión, Minimally implicit Runge-Kutta methods for Resistive Relativistic MHD, J. Phys.: Conf. Ser., 719 (2015), 2016, 012015.
[6] M. A. Aloy, J. M. Ibánez, J. M. Martí, and E. Müller, GENESIS: A high-resolution code for three-dimensional relativistic hydrodynamics, Astrophys. J. Suppl. Ser., 122 (1999), pp. 151-166.
[7] M. Anderson, E. W. Hirschmann, L. Lehner, S. L. Liebling, P. M. Motl, D. Neilsen, C. Palenzuela, and J. E. Tohline, Magnetized neutron-star mergers and gravitational-wave signals, Phys. Rev. Lett., 100 (2008), 191101.
[8] A. M. Anile, Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0701.76003
[9] A. M. Anile, J. C. Miller, and S. Motta, Formation and damping of relativistic strong shocks, Phys. Fluids, 26 (1983), pp. 1450-1460. · Zbl 0528.76127
[10] P. Anninos, P. C. Fragile, and J. D. Salmonson, Cosmos++: Relativistic magnetohydrodynamics on unstructured grids with local adaptive refinement, Astrophys. J., 635 (2005), pp. 723-740.
[11] L. Antón, O. Zanotti, J. A. Miralles, J. M. Martí, J. M. Ibán͂ez, J. A. Font, and J. A. Pons, Numerical 3+1 general relativistic magnetohydrodynamics: A local characteristic approach, Astrophys. J., 637 (2006), pp. 296-312.
[12] R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, in Gravitation: An Introduction to Current Research, John Wiley & Sons, New York, 1962, pp. 227-265. · Zbl 1152.83320
[13] R. Arnowitt, S. Deser, and C. W. Misner, Republication of: The dynamics of general relativity, Gen. Relativ. Gravit., 40 (2008), pp. 1997-2027. · Zbl 1152.83320
[14] L. Arpaia and M. Ricchiuto, Well balanced residual distribution for the ALE spherical shallow water equations on moving adaptive meshes, J. Comput. Phys., 405 (2020), 109173. · Zbl 1453.65295
[15] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065, https://doi.org/10.1137/S1064827503431090. · Zbl 1133.65308
[16] L. Baiotti, I. Hawke, P. J. Montero, F. Löffler, L. Rezzolla, N. Stergioulas, J. A. Font, and E. Seidel, Three-dimensional relativistic simulations of rotating neutron-star collapse to a Kerr black hole, Phys. Rev. D, 71 (2005), 024035.
[17] F. Banyuls, J. A. Font, J. M. Ibán͂ez, J. M. Martí, and J. A. Miralles, Numerical 3+1 general relativistic hydrodynamics: A local characteristic approach, Astrophys. J., 476 (1997), pp. 221-231.
[18] C. Bassi, L. Bonaventura, S. Busto, and M. Dumbser, A hyperbolic reformulation of the Serre-Green-Naghdi model for general bottom topographies, Comput. & Fluids, 212 (2020), 104716. · Zbl 07335348
[19] T. W. Baumgarte and S. L. Shapiro, Numerical integration of Einstein’s field equations, Phys. Rev. D, 59 (1998), 024007. · Zbl 1250.83004
[20] T. W. Baumgarte and S. L. Shapiro, Numerical Relativity: Solving Einstein’s Equations on the Computer, Cambridge University Press, 2010. · Zbl 1198.83001
[21] J. P. Berberich, P. Chandrashekar, and C. Klingenberg, High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws, Comput. & Fluids, 219 (2021), 104858. · Zbl 07426188
[22] A. Bermúdez, X. López, and M. E. Vázquez-Cendón, Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines, J. Comput. Phys., 323 (2016), pp. 126-148. · Zbl 1415.76463
[23] A. Bermudez and M. E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids, 23 (1994), pp. 1049-1071. · Zbl 0816.76052
[24] S. Bernuzzi and D. Hilditch, Constraint violation in free evolution schemes: Comparing the BSSNOK formulation with a conformal decomposition of the Z4 formulation, Phys. Rev. D, 81 (2010), 084003.
[25] C. Bona, T. Ledvinka, C. Palenzuela, and M. Zácek, General-covariant evolution formalism for numerical relativity, Phys. Rev. D, 67 (2003), 104005. · Zbl 1074.83003
[26] C. Bona, T. Ledvinka, C. Palenzuela, and M. Zácek, Symmetry-breaking mechanism for the Z4 general-covariant evolution system, Phys. Rev. D, 69 (2004), 64036. · Zbl 1074.83003
[27] C. Bona, J. Massó, E. Seidel, and J. Stela, New formalism for numerical relativity, Phys. Rev. Lett., 75 (1995), pp. 600-603.
[28] S. Bonazzola, E. Gourgoulhon, P. Grandclement, and J. Novak, Constrained scheme for the Einstein equations based on the Dirac gauge and spherical coordinates, Phys. Rev. D, 70 (2004), 104007.
[29] N. Botta, R. Klein, S. Langenberg, and S. Lützenkirchen, Well balanced finite volume methods for nearly hydrostatic flows, J. Comput. Phys., 196 (2004), pp. 539-565. · Zbl 1109.86304
[30] F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Springer Science + Business Media, 2004. · Zbl 1086.65091
[31] N. Bucciantini and L. Del Zanna, General relativistic magnetohydrodynamics in axisymmetric dynamical spacetimes: The X-ECHO code, Astron. Astrophys., 528 (2011), A101.
[32] N. Bucciantini and L. Del Zanna, A fully covariant mean-field dynamo closure for numerical 3+1 resistive GRMHD, Monthly Not. Roy. Astr. Soc., 428 (2013), pp. 71-85.
[33] M. Bugli, L. Del Zanna, and N. Bucciantini, Dynamo action in thick discs around Kerr black holes: High-order resistive GRMHD simulations, Monthly Not. Roy. Astr. Soc. Lett., 440 (2014), pp. L41-L45.
[34] M. Bugner, Discontinuous Galerkin Methods for General Relativistic Hydrodynamics, Ph.D. thesis, Friedrich-Schiller-Universität Jena, Jena, Germany, 2018.
[35] S. Busto, M. Dumbser, C. Escalante, S. Gavrilyuk, and N. Favrie, On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems, J. Sci. Comput., 87 (2021), 48. · Zbl 1465.76060
[36] S. Carroll, Spacetime and Geometry. An Introduction to General Relativity, Addison-Wesley, San Francisco, 2004. · Zbl 1131.83001
[37] M. Castro, J. Gallardo, and C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems, Math. Comp., 75 (2006), pp. 1103-1134. · Zbl 1096.65082
[38] M. Castro, J. M. Gallardo, J. A. López-García, and C. Parés, Well-balanced high order extensions of Godunov’s method for semilinear balance laws, SIAM J. Numer. Anal., 46 (2008), pp. 1012-1039, https://doi.org/10.1137/060674879. · Zbl 1159.74045
[39] M. J. Castro, T. M. de Luna, and C. Parés, Well-balanced schemes and path-conservative numerical methods, in Handbook of Numerical Methods for Hyperbolic Problems, Handb. Numer. Anal. 18, Elsevier/North-Holland, Amsterdam, 2017, pp. 131-175. · Zbl 1368.65131
[40] M. J. Castro and C. Parés, Well-balanced high-order finite volume methods for systems of balance laws, J. Sci. Comput., 82 (2020), pp. 1-48. · Zbl 1440.65109
[41] P. Chandrashekar and C. Klingenberg, A second order well-balanced finite volume scheme for Euler equations with gravity, SIAM J. Sci. Comput., 37 (2015), pp. B382-B402, https://doi.org/10.1137/140984373. · Zbl 1320.76078
[42] A. Y. Chernyshenko, M. A. Olshanskii, and Y. V. Vassilevski, A hybrid finite volume-finite element method for bulk-surface coupled problems, J. Comput. Phys., 352 (2018), pp. 516-533. · Zbl 1375.76098
[43] S. Chiocchetti, I. Peshkov, S. Gavrilyuk, and M. Dumbser, High order ADER schemes and GLM curl cleaning for a first order hyperbolic formulation of compressible flow with surface tension, J. Comput. Phys., 426 (2021), 109898.
[44] L. Cirrottola, M. Ricchiuto, A. Froehly, B. Re, A. Guardone, and G. Quaranta, Adaptive deformation of 3D unstructured meshes with curved body fitted boundaries with application to unsteady compressible flows, J. Comput. Phys., 433 (2021), 110177.
[45] I. Cordero-Carrión, P. Cerdá-Durán, H. Dimmelmeier, J. L. Jaramillo, J. Novak, and E. Gourgoulhon, Improved constrained scheme for the Einstein equations: An approach to the uniqueness issue, Phys. Rev. D, 79 (2009), 024017. · Zbl 1222.83024
[46] I. Cordero-Carrión, J. M. Ibanez, E. Gourgoulhon, J. L. Jaramillo, and J. Novak, Mathematical issues in a fully constrained formulation of the Einstein equations, Phys. Rev. D, 77 (2008), 084007.
[47] G. Dal Maso, P. G. Lefloch, and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9), 74 (1995), pp. 483-548. · Zbl 0853.35068
[48] L. Del Zanna and N. Bucciantini, An efficient shock-capturing central-type scheme for multidimensional relativistic flows. I. Hydrodynamics, Astron. Astrophys., 390 (2002), pp. 1177-1186. · Zbl 1209.76022
[49] L. Del Zanna, O. Zanotti, N. Bucciantini, and P. Londrillo, ECHO: A Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics, Astron. Astrophys., 473 (2007), pp. 11-30.
[50] V. Desveaux, M. Zenk, C. Berthon, and C. Klingenberg, A well-balanced scheme to capture non-explicit steady states in the Euler equations with gravity, Internat. J. Numer. Methods Fluids, 81 (2016), pp. 104-127. · Zbl 1382.65310
[51] K. Dionysopoulou, D. Alic, C. Palenzuela, L. Rezzolla, and B. Giacomazzo, General-relativistic resistive magnetohydrodynamics in three dimensions: Formulation and tests, Phys. Rev. D, 88 (2013), 044020.
[52] M. D. Duez, Y. T. Liu, S. L. Shapiro, and B. C. Stephens, Relativistic magnetohydrodynamics in dynamical spacetimes: Numerical methods and tests, Phys. Rev. D, 72 (2005), 024028.
[53] M. Dumbser, S. Chiocchetti, and I. Peshkov, On numerical methods for hyperbolic PDE with curl involutions, in Continuum Mechanics, Applied Mathematics and Scientific Computing: Godunov’s Legacy, Springer, 2020, pp. 125-134.
[54] M. Dumbser, F. Fambri, E. Gaburro, and A. Reinarz, On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations, J. Comput. Phys., 404 (2020), 109088. · Zbl 1453.85002
[55] M. Dumbser, F. Guercilena, S. Köppel, L. Rezzolla, and O. Zanotti, Conformal and covariant Z4 formulation of the Einstein equations: Strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes, Phys. Rev. D, 97 (2018), 084053.
[56] M. Dumbser and O. Zanotti, Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations, J. Comput. Phys., 228 (2009), pp. 6991-7006. · Zbl 1261.76028
[57] F. Fambri, M. Dumbser, S. Köppel, L. Rezzolla, and O. Zanotti, ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics, Monthly Not. Roy. Astr. Soc., 477 (2018), pp. 4543-4564.
[58] J. A. Font, Numerical hydrodynamics and magnetohydrodynamics in general relativity, Living Rev. Relativ., 11 (2008), 7. · Zbl 1166.83003
[59] E. Gaburro, A unified framework for the solution of hyperbolic PDE systems using high order direct Arbitrary-Lagrangian-Eulerian schemes on moving unstructured meshes with topology change, Arch. Comput. Methods Eng., 28 (2021), pp. 1249-1321.
[60] E. Gaburro, W. Boscheri, S. Chiocchetti, C. Klingenberg, V. Springel, and M. Dumbser, High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes, J. Comput. Phys., 407 (2020), 109167.
[61] E. Gaburro, M. J. Castro, and M. Dumbser, A well balanced diffuse interface method for complex nonhydrostatic free surface flows, Comput. & Fluids, 175 (2018), pp. 180-198. · Zbl 1410.76224
[62] E. Gaburro, M. J. Castro, and M. Dumbser, Well-balanced Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming meshes for the Euler equations of gas dynamics with gravity, Monthly Not. Roy. Astr. Soc., 477 (2018), pp. 2251-2275.
[63] E. Gaburro and M. Dumbser, A posteriori subcell finite volume limiter for general \(P_NP_M\) schemes: Applications from gasdynamics to relativistic magnetohydrodynamics, J. Sci. Comput., 86 (2021), 37. · Zbl 1475.65091
[64] E. Gaburro, M. Dumbser, and M. J. Castro, Direct Arbitrary-Lagrangian-Eulerian finite volume schemes on moving nonconforming unstructured meshes, Comput. & Fluids, 159 (2017), pp. 254-275. · Zbl 1390.76433
[65] B. Giacomazzo and L. Rezzolla, WhiskyMHD: A new numerical code for general relativistic magnetohydrodynamics, Classical Quantum Gravity, 24 (2007), pp. S235-S258. · Zbl 1117.83002
[66] L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms, Math. Models Methods Appl. Sci., 11 (2001), pp. 339-365. · Zbl 1018.65108
[67] L. Grosheintz-Laval and R. Käppeli, High-order well-balanced finite volume schemes for the Euler equations with gravitation, J. Comput. Phys., 378 (2019), pp. 324-343. · Zbl 1416.65266
[68] C. Gundlach and J. M. Martín-García, Hyperbolicity of second order in space systems of evolution equations, Classical Quantum Gravity, 23 (2006), pp. S387-S404. · Zbl 1191.83009
[69] R. Käppeli and S. Mishra, Well-balanced schemes for the Euler equations with gravitation, J. Comput. Phys., 259 (2014), pp. 199-219. · Zbl 1349.76345
[70] R. Käppeli and S. Mishra, A well-balanced finite volume scheme for the Euler equations with gravitation-The exact preservation of hydrostatic equilibrium with arbitrary entropy stratification, Astron. Astrophys., 587 (2016), A94.
[71] K. Kiuchi, Y. Sekiguchi, M. Shibata, and K. Taniguchi, Long-term general relativistic simulation of binary neutron stars collapsing to a black hole, Phys. Rev. D, 80 (2009), 064037.
[72] C. Klingenberg, G. Puppo, and M. Semplice, Arbitrary order finite volume well-balanced schemes for the Euler equations with gravity, SIAM J. Sci. Comput., 41 (2019), pp. A695-A721, https://doi.org/10.1137/18M1196704. · Zbl 1412.65125
[73] S. Komissarov, General relativistic magnetohydrodynamic simulations of monopole magnetospheres of black holes, Monthly Not. Roy. Astr. Soc., 350 (2004), pp. 1431-1436.
[74] R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., 146 (1998), pp. 346-365. · Zbl 0931.76059
[75] J. M. Martí and E. Müller, Grid-based methods in relativistic hydrodynamics and magnetohydrodynamics, Living Rev. Comput. Astrophys., 1 (2015), 3.
[76] F. C. Michel, Accretion of matter by condensed objects, Astrophys. Space Sci., 15 (1972), pp. 153-160.
[77] T. Nakamura, K. Oohara, and Y. Kojima, General relativistic collapse to black holes and gravitational waves from black holes, Progr. Theoret. Phys. Suppl., no. 90 (1987), pp. 1-218.
[78] J. R. Oppenheimer and G. M. Volkoff, On massive neutron cores, Phys. Rev., 55 (1939), pp. 374-381. · Zbl 0020.28501
[79] C. Palenzuela, L. Lehner, O. Reula, and L. Rezzolla, Beyond ideal MHD: Towards a more realistic modelling of relativistic astrophysical plasmas, Monthly Not. Roy. Astr. Soc., 394 (2009), pp. 1727-1740.
[80] C. Parés, Numerical methods for nonconservative hyperbolic systems: A theoretical framework, SIAM J. Numer. Anal., 44 (2006), pp. 300-321, https://doi.org/10.1137/050628052. · Zbl 1130.65089
[81] B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), pp. 201-231. · Zbl 1008.65066
[82] O. Porth, H. Olivares, Y. Mizuno, Z. Younsi, L. Rezzolla, M. Moscibrodzka, H. Falcke, and M. Kramer, The black hole accretion code, Comput. Astrophys. Cosmol., 4 (2017), pp. 1-42.
[83] D. Radice and L. Rezzolla, THC: A new high-order finite-difference high-resolution shock-capturing code for special-relativistic hydrodynamics, Astron. Astrophys., 547 (2012), A26.
[84] D. Radice, L. Rezzolla, and F. Galeazzi, Beyond second-order convergence in simulations of binary neutron stars in full general relativity, Monthly Not. Roy. Astr. Soc. Lett., 437 (2013), pp. L46-L50.
[85] T. C. Rebollo, A. D. Delgado, and E. D. F. Nieto, A family of stable numerical solvers for the shallow water equations with source terms, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 203-225. · Zbl 1083.76557
[86] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics, Oxford University Press, 2013.
[87] G. Russo and A. Anile, Stability properties of relativistic shock waves: Basic results, Phys. Fluids, 30 (1987), pp. 2406-2413. · Zbl 0663.76143
[88] K. Schaal, A. Bauer, P. Chandrashekar, R. Pakmor, C. Klingenberg, and V. Springel, Astrophysical hydrodynamics with a high-order discontinuous Galerkin scheme and adaptive mesh refinement, Monthly Not. Roy. Astr. Soc., 453 (2015), pp. 4278-4300.
[89] M. Shibata and T. Nakamura, Evolution of three-dimensional gravitational waves: Harmonic slicing case, Phys. Rev. D, 52 (1995), pp. 5428-5444. · Zbl 1250.83027
[90] V. Springel, E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh, Monthly Not. Roy. Astr. Soc., 401 (2010), pp. 791-851.
[91] R. Takahashi and M. Umemura, General relativistic radiative transfer code in rotating black hole space-time: ARTIST, Monthly Not. Roy. Astr. Soc., 464 (2016), pp. 4567-4585.
[92] H. Tang, T. Tang, and K. Xu, A gas-kinetic scheme for shallow-water equations with source terms, Z. Angew. Math. Phys., 55 (2004), pp. 365-382. · Zbl 1142.76457
[93] A. Thomann, M. Zenk, and C. Klingenberg, A second-order positivity-preserving well-balanced finite volume scheme for Euler equations with gravity for arbitrary hydrostatic equilibria, Internat. J. Numer. Methods Fluids, 89 (2019), pp. 465-482.
[94] R. C. Tolman, Static solutions of Einstein’s field equations for spheres of fluid, Phys. Rev., 55 (1939), pp. 364-373. · JFM 65.1048.02
[95] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd ed., Springer, 1999. · Zbl 0923.76004
[96] B. Van Leer, Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme, J. Comput. Phys., 14 (1974), pp. 361-370. · Zbl 0276.65055
[97] B. Van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., 32 (1979), pp. 101-136. · Zbl 1364.65223
[98] R. M. Wald, General Relativity, University of Chicago Press, Chicago, 1984. · Zbl 0549.53001
[99] C. J. White, J. M. Stone, and C. F. Gammie, An extension of the Athena++ code framework for GRMHD based on advanced Riemann solvers and staggered-mesh constrained transport, Astrophys. J. Suppl. Ser., 225 (2016), 22.
[100] J. R. Wilson, Some Magnetic Effects in Stellar Collapse and Accretion, Tech. report, Lawrence Livermore Laboratory, University of California, Livermore, CA, 1975.
[101] Y. Xing, Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium, J. Comput. Phys., 257 (2014), pp. 536-553. · Zbl 1349.76289
[102] Y. Xing and C.-W. Shu, High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, J. Comput. Phys., 214 (2006), pp. 567-598. · Zbl 1089.65091
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.