A Well Balanced Finite Volume Scheme for General Relativity
Abstract
In this work we present a novel second order accurate well balanced (WB) finite volume (FV) scheme for the solution of the general relativistic magnetohydrodynamics (GRMHD) equations and the first order CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, as well as the fully coupled FO-CCZ4 + GRMHD system. These systems of first order hyperbolic PDEs allow one to study the dynamics of the matter and the dynamics of the space-time according to the theory of general relativity. The new well balanced finite volume scheme presented here exploits the knowledge of an equilibrium solution of interest when integrating the conservative fluxes, the nonconservative products, and the algebraic source terms, and also when performing the piecewise linear data reconstruction. This results in a rather simple modification of the underlying second order FV scheme, which, however, being able to cancel numerical errors committed w.r.t. the equilibrium component of the numerical solution, substantially improves the accuracy and long-time stability of the numerical scheme when simulating small perturbations of stationary equilibria. In particular, the need for well balanced techniques appears to be more and more crucial as the applications increase their complexity. We close the paper with a series of numerical tests of increasing difficulty, where we study the evolution of small perturbations of accretion problems and of stable Tolman--Oppenheimer--Volkoff (TOV) neutron stars. Our results show that the well balancing significantly improves the long-time stability of the finite volume scheme compared to a standard scheme.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
M. Alcubierre, Introduction to 3+1 Numerical Relativity, Internat. Ser. Mongr. Phys. 140, Oxford University Press, 2008.
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.
9.
A. M. Anile, J. C. Miller, and S. Motta, Formation and damping of relativistic strong shocks, Phys. Fluids, 26 (1983), pp. 1450--1460.
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.
13.
R. Arnowitt, S. Deser, and C. W. Misner, Republication of: The dynamics of general relativity, Gen. Relativ. Gravit., 40 (2008), pp. 1997--2027.
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.
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.
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.
19.
T. W. Baumgarte and S. L. Shapiro, Numerical integration of Einstein's field equations, Phys. Rev. D, 59 (1998), 024007.
20.
T. W. Baumgarte and S. L. Shapiro, Numerical Relativity: Solving Einstein's Equations on the Computer, Cambridge University Press, 2010.
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.
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.
23.
A. Bermudez and M. E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids, 23 (1994), pp. 1049--1071.
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.
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.
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.
30.
F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Springer Science + Business Media, 2004.
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.
36.
S. Carroll, Spacetime and Geometry. An Introduction to General Relativity, Addison-Wesley, San Francisco, 2004.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
65.
B. Giacomazzo and L. Rezzolla, WhiskyMHD: A new numerical code for general relativistic magnetohydrodynamics, Classical Quantum Gravity, 24 (2007), pp. S235--S258.
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.
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.
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.
69.
R. Käppeli and S. Mishra, Well-balanced schemes for the Euler equations with gravitation, J. Comput. Phys., 259 (2014), pp. 199--219.
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.
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.
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.
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.
81.
B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), pp. 201--231.
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.
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.
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.
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.
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.
95.
E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd ed., Springer, 1999.
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.
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.
98.
R. M. Wald, General Relativity, University of Chicago Press, Chicago, 1984.
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.
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.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: B1226 - B1251
DOI: 10.1137/21M1399154
ISSN (online): 1095-7197
Copyright
© 2021, Society for Industrial and Applied Mathematics.
History
Submitted: 16 February 2021
Accepted: 30 August 2021
Published online: 20 December 2021
Keywords
MSC codes
Authors
Funding Information
ERDF : RTI2018-096064-B-C21, UMA18-Federja-161, P18-RT-3163
University of Trento
Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 671698
Horizon 2020 Framework Programme https://doi.org/10.13039/100010661 : 101025563
Istituto Nazionale di Alta Matematica "Francesco Severi" https://doi.org/10.13039/100009112
Ministero dell'Istruzione, dell'Università e della Ricerca https://doi.org/10.13039/501100003407
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Discontinuous Galerkin schemes for hyperbolic systems in non-conservative variables: Quasi-conservative formulation with subcell finite volume correctionsComputer Methods in Applied Mechanics and Engineering, Vol. 431 | 1 Nov 2024
- Well-balanced High-order Finite Difference Weighted Essentially Nonoscillatory Schemes for a First-order Z4 Formulation of the Einstein Field EquationsThe Astrophysical Journal Supplement Series, Vol. 275, No. 1 | 8 November 2024
- High-order discontinuous Galerkin schemes with subcell finite volume limiter and adaptive mesh refinement for a monolithic first-order BSSNOK formulation of the Einstein-Euler equationsPhysical Review D, Vol. 110, No. 8 | 3 October 2024
- A well-balanced discontinuous Galerkin method for the first–order Z4 formulation of the Einstein–Euler systemJournal of Computational Physics, Vol. 504 | 1 May 2024
- A Well-Balanced Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at All Mach NumbersJournal of Scientific Computing, Vol. 98, No. 2 | 3 January 2024
- A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equationsJournal of Computational Physics, Vol. 493 | 1 Nov 2023
- Well balanced finite volume schemes for shallow water equations on manifoldsApplied Mathematics and Computation, Vol. 441 | 1 Mar 2023
- High-Order Arbitrary-Lagrangian-Eulerian Schemes on Crazy Moving Voronoi MeshesAdvances in Numerical Methods for Hyperbolic Balance Laws and Related Problems | 3 June 2023
- Well-balanced methods for computational astrophysicsLiving Reviews in Computational Astrophysics, Vol. 8, No. 1 | 19 October 2022
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].