On the convergence of the discontinuous Galerkin scheme for Einstein-scalar equations

Chen, Yuewen; Shu, Chi-Wang; Yau, Shing-Tung

doi:10.1090/mcom/4077

On the convergence of the discontinuous Galerkin scheme for Einstein-scalar equations
HTML articles powered by AMS MathViewer

by Yuewen Chen, Chi-Wang Shu and Shing-Tung Yau;

Math. Comp.

DOI: https://doi.org/10.1090/mcom/4077

Published electronically: February 28, 2025

PDF | Request permission

Supplementary Appendix: Appendix A

Abstract:

We prove the stability and convergence of the high order discontinuous Galerkin scheme to spherically symmetric Einstein-scalar equations for a class of large initial data that ensures the formation of a black hole. Having chosen the Bondi coordinate system, we achieve $L^2$ stability and obtain the optimal error estimates.

References

Peter Anninos, Karen Camarda, Joan Massó, Edward Seidel, Wai-Mo Suen, and John Towns, Three-dimensional numerical relativity: the evolution of black holes, Phys. Rev. D (3) 52 (1995), no. 4, 2059–2082. MR 1345767, DOI 10.1103/PhysRevD.52.2059
R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, Gravitation: An introduction to current research, Wiley, New York-London, 1962, pp. 227–265. MR 143629
J. G. Baker, J. Centrella, D. I. Choi, M. Koppitz, and J. van Meter, Gravitational-wave extraction from an inspiraling configuration of merging black holes, Phys. Rev. Lett. 96 (2006), 111102.
Robert Bartnik, Einstein equations in the null quasispherical gauge, Classical Quantum Gravity 14 (1997), no. 8, 2185–2194. MR 1468576, DOI 10.1088/0264-9381/14/8/017
Thomas W. Baumgarte and Stuart L. Shapiro, Numerical integration of Einstein’s field equations, Phys. Rev. D (3) 59 (1999), no. 2, 024007, 7. MR 1692065, DOI 10.1103/PhysRevD.59.024007
Nigel T. Bishop, Roberto Gómez, Luis Lehner, Manoj Maharaj, and Jeffrey Winicour, High-powered gravitational news, Phys. Rev. D (3) 56 (1997), no. 10, 6298–6309. MR 1487038, DOI 10.1103/PhysRevD.56.6298
H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems, Proc. Roy. Soc. London Ser. A 269 (1962), 21–52. MR 147276, DOI 10.1098/rspa.1962.0161
Bernd Brügmann, Binary black hole mergers in 3d numerical relativity, Internat. J. Modern Phys. D 8 (1999), no. 1, 85–100. MR 1675728, DOI 10.1142/S0218271899000080
B. Brügmann, J. A. González, M. Hannam, S. Husa, U. Sperhake, and W. Tichy, Calibration of moving puncture simulations, Phys. Rev. D 77 (2008), 024027.
M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Accurate evolutions of orbiting black-hole binaries without excision, Phys. Rev. Lett. 96 (2006), 111101.
Z. Cao, H. J. Yo, and J. P. Yu, Reinvestigation of moving punctured black holes with a new code, Phys. Rev. D 78 (2008), 124011.
Demetrios Christodoulou, The problem of a self-gravitating scalar field, Comm. Math. Phys. 105 (1986), no. 3, 337–361. MR 848643
Demetrios Christodoulou, Global existence of generalized solutions of the spherically symmetric Einstein-scalar equations in the large, Comm. Math. Phys. 106 (1986), no. 4, 587–621. MR 860312
Demetrios Christodoulou, A mathematical theory of gravitational collapse, Comm. Math. Phys. 109 (1987), no. 4, 613–647. MR 885564
Demetrios Christodoulou, The structure and uniqueness of generalized solutions of the spherically symmetric Einstein-scalar equations, Comm. Math. Phys. 109 (1987), no. 4, 591–611. MR 885563
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Bernardo Cockburn, Suchung Hou, and Chi-Wang Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp. 54 (1990), no. 190, 545–581. MR 1010597, DOI 10.1090/S0025-5718-1990-1010597-0
Bernardo Cockburn, San Yih Lin, and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems, J. Comput. Phys. 84 (1989), no. 1, 90–113. MR 1015355, DOI 10.1016/0021-9991(89)90183-6
Bernardo Cockburn and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), no. 186, 411–435. MR 983311, DOI 10.1090/S0025-5718-1989-0983311-4
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 3, 337–361 (English, with French summary). MR 1103092, DOI 10.1051/m2an/1991250303371
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224. MR 1619652, DOI 10.1006/jcph.1998.5892
N. Deppe, Binary neutron star mergers using a discontinuous Galerkin-finite difference hybrid method, arXiv:2406.19038v1.
N. Deppe, Simulating magnetized neutron stars with discontinuous Galerkin methods, arXiv:2109.12033.
H. P. de Oliveira, E. L. Rodrigues, and J. E. F. Skea, Gravitational collapse of scalar fields via spectral methods, Phys. Rev. D 82 (2010), 104023.
R. A. d’Inverno, M. R. Dubal, and E. A. Sarkies, Cauchy-characteristic matching for a family of cylindrical solutions possessing both gravitational degrees of freedom, Classical Quantum Gravity 17 (2000), no. 16, 3157–3170. MR 1779505, DOI 10.1088/0264-9381/17/16/305
K. Eppley, The numerical evolution of the collision of two black holes, Dissertation for Doctoral Degree, Princeton University, Pricneton, 1975.
G. Lovelace, Simulating binary black hole mergers using discontinuous Galerkin methods, arXiv:2410.00265.
D. Garfinkle, Choptuik scaling in null coordinates, Phys. Rev. D 51 (1995), 5558.
R. Gómez, R. A. Isaacson, J. S. Welling, and J. Winicour, Numerical relativity on null cones, Dynamical spacetimes and numerical relativity (Philadelphia, PA, 1985) Cambridge Univ. Press, Cambridge, 1986, pp. 236–255. MR 945489
R. Gómez, L. Lehner, R. L. Marsa, and J. Winicour, Moving black holes in 3D, Phys. Rev. D (3) 57 (1998), no. 8, 4778–4788. MR 1682357, DOI 10.1103/PhysRevD.57.4778
R. Gomez, L. Lehner, R. L. Marsa, et al., Stable characteristic evolution of generic three-dimensional single-black-hole spacetimes, Phys. Rev. Lett. 80 (1998), 3915–3918.
Susan G. Hahn and Richard W. Lindquist, The two-body problem in geometrodynamics, Ann. Physics 29 (1964), 304–331. MR 177779, DOI 10.1016/0003-4916(64)90223-4
Frank Herrmann, Ian Hinder, Deirdre Shoemaker, and Pablo Laguna, Unequal mass binary black hole plunges and gravitational recoil, Classical Quantum Gravity 24 (2007), no. 12, S33–S42. MR 2333150, DOI 10.1088/0264-9381/24/12/S04
Guang Shan Jiang and Chi-Wang Shu, On a cell entropy inequality for discontinuous Galerkin methods, Math. Comp. 62 (1994), no. 206, 531–538. MR 1223232, DOI 10.1090/S0025-5718-1994-1223232-7
C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp. 46 (1986), no. 173, 1–26. MR 815828, DOI 10.1090/S0025-5718-1986-0815828-4
Lawrence E. Kidder, Mark A. Scheel, Saul A. Teukolsky, Eric D. Carlson, and Gregory B. Cook, Black hole evolution by spectral methods, Phys. Rev. D (3) 62 (2000), no. 8, 084032, 20. MR 1804090, DOI 10.1103/PhysRevD.62.084032
M. Koppitz, D. Pollney, C. Reisswig, L. Rezzolla, J. Thornburg, P. Diener, and E. Schnetter, Recoil velocities from equal-mass binary-black-hole mergers, Phys. Rev. Lett. 99 (2007), 041102.
P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Academic Press, New York-London, 1974, pp. 89–123. MR 658142
C. W. Misner, K. S. Thorne, and W. H. Zurek, John Wheeler, relativity, and quantum information, Phys. Today 62 (2009), 40–46.
Todd E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal. 28 (1991), no. 1, 133–140. MR 1083327, DOI 10.1137/0728006
T. Piran, Cylindrical general relativistic collapse, Phys. Rev. Lett. 41 (1978), 1085.
Frans Pretorius, Evolution of binary black-hole spacetimes, Phys. Rev. Lett. 95 (2005), no. 12, 121101, 4. MR 2169088, DOI 10.1103/PhysRevLett.95.121101
Frans Pretorius, Numerical relativity using a generalized harmonic decomposition, Classical Quantum Gravity 22 (2005), no. 2, 425–451. MR 2119674, DOI 10.1088/0264-9381/22/2/014
O. Rinne, Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations, Classical Quantum Gravity 23 (2006), no. 22, 6275–6300. MR 2272007, DOI 10.1088/0264-9381/23/22/013
Milton Ruiz, Oliver Rinne, and Olivier Sarbach, Outer boundary conditions for Einstein’s field equations in harmonic coordinates, Classical Quantum Gravity 24 (2007), no. 24, 6349–6377. MR 2374549, DOI 10.1088/0264-9381/24/24/012
Gerard R. Richter, An optimal-order error estimate for the discontinuous Galerkin method, Math. Comp. 50 (1988), no. 181, 75–88. MR 917819, DOI 10.1090/S0025-5718-1988-0917819-3
W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
Chi-Wang Shu, TVB uniformly high-order schemes for conservation laws, Math. Comp. 49 (1987), no. 179, 105–121. MR 890256, DOI 10.1090/S0025-5718-1987-0890256-5
Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471. MR 954915, DOI 10.1016/0021-9991(88)90177-5
Masaru Shibata and Takashi Nakamura, Evolution of three-dimensional gravitational waves: harmonic slicing case, Phys. Rev. D (3) 52 (1995), no. 10, 5428–5444. MR 1360417, DOI 10.1103/PhysRevD.52.5428
F. Siebel, J. A. Font, E. Muller, and P. Papadopoulos, Simulating the dynamics of relativistic stars via a light-cone approach, Phys. Rev. D 65 (2002), 064038.
L. Smarr, The structure of general relativity with a numerical example, Dissertation for Doctoral Degree, University of Texas at Austin, Austin, 1975.
R. F. Stark and T. Piran, Gravitational-wave emission from rotating gravitational collapse, Phys. Rev. Lett. 56 (1986), 97.
Béla Szilágyi, Lee Lindblom, and Mark A. Scheel, Simulations of binary black hole mergers using spectral methods, Phys. Rev. D 80 (2009), no. 12, 124010, 17. MR 2669798, DOI 10.1103/PhysRevD.80.124010
John Archibald Wheeler, Geometrodynamics and the issue of the final state, Relativité, Groupes et Topologie (Lectures, Les Houches, 1963 Summer School of Theoret. Phys., Univ. Grenoble), Gordon and Breach, New York-London, 1964, pp. 315–520. MR 168332
T. Yamamoto, M. Shibata, and K. Taniguchi, Simulating coalescing compact binaries by a new code (SACRA), Phys. Rev. D 78 (2008), 064054.
Qiang Zhang and Chi-Wang Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004), no. 2, 641–666. MR 2084230, DOI 10.1137/S0036142902404182
Qiang Zhang and Chi-Wang Shu, Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizable systems of conservation laws, SIAM J. Numer. Anal. 44 (2006), no. 4, 1703–1720. MR 2257123, DOI 10.1137/040620382

AMS Website Down

Mathematics of Computation

On the convergence of the discontinuous Galerkin scheme for Einstein-scalar equations
HTML articles powered by AMS MathViewer

Abstract: