Characteristic evolution and matching.

*(English)*Zbl 1166.83300Summary: I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to eliminate the role of this artificial outer boundary via Cauchy-characteristic matching, by which the radiated waveform can be computed at null infinity. Progress in this direction is discussed.

Update of the author’s paper [Zbl 1316.83015], see also the updates [Zbl 1023.83004; Zbl 1316.83016]: Significantly updated previous version. About 30 references have been added. The text has been revised at several points. Added new Sections 4, 5.3 and 6.

Update of the author’s paper [Zbl 1316.83015], see also the updates [Zbl 1023.83004; Zbl 1316.83016]: Significantly updated previous version. About 30 references have been added. The text has been revised at several points. Added new Sections 4, 5.3 and 6.

##### MSC:

83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |

83-08 | Computational methods for problems pertaining to relativity and gravitational theory |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

##### Software:

AHFinderDirect##### References:

[1] | Abrahams, AM; Evans, CR, Gauge-invariant treatment of gravitational radiation near the source: analysis and numerical simulations, Phys. Rev. D, 42, 2585-2594, (1990) |

[2] | Abrahams, AM; Price, RH, Applying black hole perturbation theory to numerically generated spacetimes, Phys. Rev. D, 53, 1963-1971, (1996) |

[3] | Abrahams, AM; Rezzolla, L; Rupright, ME; Anderson, A; Anninos, P; Baumgarte, TW; Bishop, NT; Brandt, SR; Browne, JC; Camarda, K; Choptuik, MW; Cook, GB; Correll, RR; Evans, CR; Finn, LS; Fox, GC; Gómez, R; Haupt, T; Huq, MF; Kidder, LE; Klasky, SA; Laguna, P; Landry, W; Lehner, L; Lenaghan, J; Marsa, RL; Massó, J; Matzner, RA; Mitra, S; Papadopoulos, P; Parashar, M; Saied, F; Saylor, PE; Scheel, MA; Seidel, E; Shapiro, SL; Shoemaker, DM; Smarr, LL; Szilágyi, B; Teukolsky, SA; Putten, MHPM; Walker, P; Winicour, J; York, JW; The Binary Black Hole Grand Challenge Alliance, Gravitational wave extraction and outer boundary conditions by perturbative matching, Phys. Rev. Lett., 80, 1812-1815, (1998) |

[4] | Abrahams, AM; Shapiro, SL; Teukolsky, SA, Calculation of gravitational waveforms from black hole collisions and disk collapse: applying perturbation theory to numerical spacetimes, Phys. Rev. D, 51, 4295-4301, (1995) |

[5] | Alcubierre, M; Allen, G; Bona, C; Fiske, D; Goodale, T; Guzmán, FS; Hawke, I; Hawley, SH; Husa, S; Koppitz, M; Lechner, C; Pollney, D; Rideout, D; Salgado, M; Schnetter, E; Seidel, E; Shinkai, H-a; Shoemaker, D; Szilágyi, B; Takahashi, R; Winicour, J, Towards standard testbeds for numerical relativity, Class. Quantum Grav., 21, 589-613, (2004) · Zbl 1045.83005 |

[6] | Anderson, JL, Gravitational radiation damping in systems with compact components, Phys. Rev. D, 36, 2301-2313, (1987) · Zbl 0625.76026 |

[7] | Anderson, JL; Hobill, DW; Centrella, JM (ed.), Matched analytic-numerical solutions of wave equations, Proceedings of the Workshop, Drexel University, October 7-11, 1985, Cambridge, New York |

[8] | Anderson, JL; Hobill, DW, Mixed analytic-numerical solutions for a simple radiating system, Gen. Relativ. Gravit., 19, 563-580, (1987) |

[9] | Anderson, JL; Hobill, DW, A study of nonlinear radiation damping by matching analytic and numerical solutions, J. Comput. Phys., 75, 283-299, (1988) · Zbl 0639.65071 |

[10] | Anderson, JL; Kates, RE; Kegeles, LS; Madonna, RG, Divergent integrals of post-Newtonian gravity: nonanalytic terms in the near-zone expansion of a gravitationally radiating system found by matching, Phys. Rev. D, 25, 2038-2048, (1982) |

[11] | Anninos, P; Daues, G; Massó, J; Seidel, E; Suen, W-M, Horizon boundary conditions for black hole spacetimes, Phys. Rev. D, 51, 5562-5578, (1995) · Zbl 1445.62096 |

[12] | Arnowitt, R; Deser, S; Misner, CW; Witten, L (ed.), The dynamics of general relativity, 227-265, (1962), New York |

[13] | Babiuc, M; Szilágyi, B; Hawke, I; Zlochower, Y, Gravitational wave extraction based on Cauchy-characteristic extraction and characteristic evolution, Class. Quantum Grav., 22, 5089-5107, (2005) · Zbl 1092.83005 |

[14] | Babiuc, M.C., Bishop, N.T., Szilágyi, B., and Winicour, J., “Strategies for the characteristic extraction of gravitational waveforms”, arXiv e-print, (2008). [arXiv:0808.0861]. 4.2.1, 6, 6, 6 |

[15] | Babiuc, MC; Husa, S; Alic, D; Hinder, I; Lechner, C; Schnetter, E; Szilágyi, B; Zlochower, Y; Dorband, N; Pollney, D; Winicour, J, Implementation of standard testbeds for numerical relativity, Class. Quantum Grav., 25, 1-38, (2008) · Zbl 1144.83002 |

[16] | Babiuc, MC; Kreiss, H-O; Winicour, J, Constraint-preserving Sommerfeld conditions for the harmonic Einstein equations, Phys. Rev. D, 75, 1-13, (2007) |

[17] | Baker, J; Campanelli, M; Lousto, CO; Takahashi, R, Modeling gravitational radiation from coalescing binary black holes, Phys. Rev. D, 65, 124012, (2002) |

[18] | Baker, JG; Centrella, J; Choi, D-I; Koppitz, M; Meter, J, Binary black hole merger dynamics and waveforms, Phys. Rev. D, 73, 1-11, (2006) |

[19] | Baker, JG; Centrella, J; Choi, D-I; Koppitz, M; Meter, J, Gravitational-wave extraction from an inspiraling configuration of merging black holes, Phys. Rev. Lett., 96, 111102, (2006) |

[20] | Barreto, W; Silva, A, Gravitational collapse of a charged and radiating fluid ball in the diffusion limit, Gen. Relativ. Gravit., 28, 735-747, (1996) · Zbl 0849.53065 |

[21] | Barreto, W; Silva, A, Self-similar and charged spheres in the diffusion approximation, Class. Quantum Grav., 16, 1783-1792, (1999) · Zbl 0942.83033 |

[22] | Barreto, W; Silva, A; Gómez, R; Lehner, L; Rosales, L; Winicour, J, Three-dimensional Einstein-Klein-Gordon system in characteristic numerical relativity, Phys. Rev. D, 71, 1-12, (2005) |

[23] | Barreto, W; Gómez, R; Lehner, L; Winicour, J, Gravitational instability of a kink, Phys. Rev. D, 54, 3834-3839, (1996) · Zbl 1254.81024 |

[24] | Barreto, W; Peralta, C; Rosales, L, Equation of state and transport processes in self-similar spheres, Phys. Rev. D, 59, 1-4, (1998) |

[25] | Bartnik, R, Einstein equations in the null quasispherical gauge, Class. Quantum Grav., 14, 2185-2194, (1997) · Zbl 0877.53059 |

[26] | Bartnik, R, Shear-free null quasi-spherical space-times, J. Math. Phys., 38, 5774-5791, (1997) · Zbl 0910.53063 |

[27] | Bartnik, R; Wit, D (ed.); Bracken, AJ (ed.); Gould, MD (ed.); Pearce, PA (ed.), Interaction of gravitational waves with a black hole, Proceedings of the Congress, University of Queensland, Brisbane, Australia, July 1997, Somerville · Zbl 1253.83017 |

[28] | Bartnik, R; Weinstein, G (ed.); Weikard, R (ed.), Assessing accuracy in a numerical Einstein solver, Proceedings of an international conference, University of Alabama in Birmingham, March 16-20, 1999, Providence · Zbl 1056.83500 |

[29] | Bartnik, R; Norton, AH; Noye, BJ (ed.); Teubner, MD (ed.); Gill, AW (ed.), Numerical solution of the Einstein equations, 91, (1998), Singapore · Zbl 0981.83003 |

[30] | Bartnik, R; Norton, AH, Numerical methods for the Einstein equations in null quasi-spherical coordinates, SIAM J. Sci. Comput., 22, 917-950, (2000) · Zbl 0973.83004 |

[31] | Bartnik, R; Norton, AH; Friedrich, H (ed.); Frauendiener, J (ed.), Numerical experiments at null infinity, Proceedings of the international workshop, Tübingen, Germany, 2-4 April 2001, Berlin; New York · Zbl 1042.83003 |

[32] | Baumgarte, TW; Shapiro, SL, Numerical integration of einstein’s field equations, Phys. Rev. D, 59, 1-7, (1998) |

[33] | Baumgarte, TW; Shapiro, SL; Teukolsky, SA, Computing supernova collapse to neutron stars and black holes, Astrophys. J., 443, 717-734, (1995) |

[34] | Baumgarte, TW; Shapiro, SL; Teukolsky, SA, Computing the delayed collapse of hot neutron stars to black holes, Astrophys. J., 458, 680-691, (1996) |

[35] | Bayliss, A; Turkel, E, Radiation boundary conditions for wavelike equations, Commun. Pure Appl. Math., 33, 707-725, (1980) · Zbl 0438.35043 |

[36] | Berger, B.K., “Numerical Approaches to Spacetime Singularities”, Living Rev. Relativity, 5, lrr-2002-1, (2002). URL (cited on 20 July 2005): http://www.livingreviews.org/lrr-2002-1. 3.1 |

[37] | Bičák, J; Reilly, P; Winicour, J, Boost-rotation symmetric gravitational null cone data, Gen. Relativ. Gravit., 20, 171-181, (1988) |

[38] | Bishop, NT; d’Inverno, RA (ed.), Some aspects of the characteristic initial value problem in numerical relativity, Proceedings of the International Workshop on Numerical Relativity, Southampton, December 1991, Cambridge; New York |

[39] | Bishop, NT, Numerical relativity: combining the Cauchy and characteristic initial value problems, Class. Quantum Grav., 10, 333-341, (1993) · Zbl 0796.65145 |

[40] | Bishop, NT, Linearized solutions of the Einstein equations within a Bondi-Sachs framework, and implications for boundary conditions in numerical simulations, Class. Quantum Grav., 22, 2393-2406, (2005) · Zbl 1077.83013 |

[41] | Bishop, NT; Deshingkar, SS, New approach to calculating the news, Phys. Rev. D, 68, 1-6, (2003) |

[42] | Bishop, NT; Gómez, R; Holvorcem, PR; Matzner, RA; Papadopoulos, P; Winicour, J, Cauchy-characteristic matching: A new approach to radiation boundary conditions, Phys. Rev. Lett., 76, 4303-4306, (1996) · Zbl 0955.83503 |

[43] | Bishop, NT; Gómez, R; Holvorcem, PR; Matzner, RA; Papadopoulos, P; Winicour, J, Cauchy-characteristic evolution and waveforms, J. Comput. Phys., 136, 140-167, (1997) · Zbl 0896.65062 |

[44] | Bishop, NT; Gómez, R; Husa, S; Lehner, L; Winicour, J, Numerical relativistic model of a massive particle in orbit near a Schwarzschild black hole, Phys. Rev. D, 68, 1-12, (2003) |

[45] | Bishop, NT; Gómez, R; Isaacson, RA; Lehner, L; Szilágyi, B; Winicour, J; Bhawal, B (ed.); Iyer, BR (ed.), Cauchy-characteristic matching, 383-408, (1999), Dordrecht; Boston · Zbl 0960.83004 |

[46] | Bishop, NT; Gómez, R; Lehner, L; Maharaj, M; Winicour, J, High-powered gravitational news, Phys. Rev. D, 56, 6298-6309, (1997) |

[47] | Bishop, NT; Gómez, R; Lehner, L; Maharaj, M; Winicour, J, The incorporation of matter into characteristic numerical relativity, Phys. Rev. D, 60, 1-11, (1999) |

[48] | Bishop, NT; Gómez, R; Lehner, L; Maharaj, M; Winicour, J, Characteristic initial data for a star orbiting a black hole, Phys. Rev. D, 72, 1-16, (2005) |

[49] | Bishop, NT; Gómez, R; Lehner, L; Winicour, J, Cauchy-characteristic extraction in numerical relativity, Phys. Rev. D, 54, 6153-6165, (1996) |

[50] | Bishop, NT; Venter, LR, Kerr metric in Bondi-Sachs form, Phys. Rev. D, 73, 1-6, (2006) |

[51] | Bizoń, P, Equivariant self-similar wave maps from Minkowski spacetime into 3-sphere, Commun. Math. Phys., 215, 45-56, (2000) · Zbl 0992.53050 |

[52] | Blaschak, JG; Kriegsmann, GA, A comparative study of absorbing boundary conditions, J. Comput. Phys., 77, 109-139, (1988) · Zbl 0655.65134 |

[53] | Bondi, H, Gravitational waves in general relativity, Nature, 186, 535-535, (1960) · Zbl 0087.42504 |

[54] | Bondi, H; Burg, MGJ; Metzner, AWK, Gravitational waves in general relativity. VII. waves from axi-symmetric isolated systems, Proc. R. Soc. London, Ser. A, 269, 21-52, (1962) · Zbl 0106.41903 |

[55] | Brady, PR; Chambers, CM; Gonçalves, SMCV, Phases of massive scalar field collapse, Phys. Rev. D, 56, r6057-r6061, (1997) |

[56] | Brady, PR; Chambers, CM; Krivan, W; Laguna, P, Telling tails in the presence of a cosmological constant, Phys. Rev. D, 55, 7538-7545, (1997) · Zbl 1328.76069 |

[57] | Brady, PR; Smith, JD, Black hole singularities: A numerical approach, Phys. Rev. Lett., 75, 1256-1259, (1995) · Zbl 1020.83590 |

[58] | Browning, GL; Hack, JJ; Swarztrauber, PN, A comparison of three numerical methods for solving differential equations on the sphere, Mon. Weather Rev., 117, 1058-1075, (1989) |

[59] | Burke, WL, Gravitational radiation damping of slowly moving systems calculated using matched asymptotic expansions, J. Math. Phys., 12, 401-418, (1971) |

[60] | Burko, LM, Structure of the black hole’s Cauchy-horizon singularity, Phys. Rev. Lett., 79, 4958-4961, (1997) · Zbl 0953.83021 |

[61] | Burko, LM; Ori, A, Late-time evolution of nonlinear gravitational collapse, Phys. Rev. D, 56, 7820-7832, (1997) |

[62] | Butler, DS, The numerical solution of hyperbolic systems of partial differential equations in three independent variables, Proc. R. Soc. London, Ser. A, 255, 232-252, (1960) · Zbl 0099.41501 |

[63] | Calabrese, G; Lehner, L; Tiglio, M, Constraint-preserving boundary conditions in numerical relativity, Phys. Rev. D, 65, 1-13, (2002) |

[64] | Calabrese, G; Pullin, J; Reula, O; Sarbach, O; Tiglio, M, Well posed constraint-preserving boundary conditions for the linearized Einstein equations, Commun. Math. Phys., 240, 377-395, (2003) · Zbl 1038.35138 |

[65] | Campanelli, M; Gómez, R; Husa, S; Winicour, J; Zlochower, Y, Close limit from a null point of view: the advanced solution, Phys. Rev. D, 63, 1-15, (2001) |

[66] | Campanelli, M; Lousto, CO; Marronetti, P; Zlochower, Y, Accurate evolutions of orbiting black-hole binaries without excision, Phys. Rev. Lett., 96, 111101, (2006) |

[67] | Choptuik, MW; d’Inverno, RA (ed.), ‘critical’ behavior in massless scalar field collapse, Proceedings of the International Workshop on Numerical Relativity, Southampton, December 1991, Cambridge; New York |

[68] | Choptuik, MW, Universality and scaling in gravitational collapse of a massless scalar field, Phys. Rev. Lett., 70, 9-12, (1993) |

[69] | Christodoulou, D, A mathematical theory of gravitational collapse, Commun. Math. Phys., 109, 613-647, (1987) · Zbl 0613.53049 |

[70] | Christodoulou, D, The formation of black holes and singularities in spherically symmetric gravitational collapse, Commun. Pure Appl. Math., 44, 339-373, (1991) · Zbl 0728.53061 |

[71] | Christodoulou, D, Bounded variation solutions of the spherically symmetric Einstein-scalar field equations, Commun. Pure Appl. Math., 46, 1131-1220, (1993) · Zbl 0853.35122 |

[72] | Christodoulou, D, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. Math. (2), 140, 607-653, (1994) · Zbl 0822.53066 |

[73] | Christodoulou, D, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. Math. (2), 149, 183-217, (1999) · Zbl 1126.83305 |

[74] | Christodoulou, D, On the global initial value problem and the issue of singularities, Class. Quantum Grav., 16, a23-a35, (1999) · Zbl 0955.83001 |

[75] | Christodoulou, D., and Klainerman, S., The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series, vol. 41, (Princeton University Press, Princeton, 1993). 3.3 · Zbl 0827.53055 |

[76] | Clarke, CJS; d’Inverno, RA, Combining Cauchy and characteristic numerical evolutions in curved coordinates, Class. Quantum Grav., 11, 1463-1448, (1994) |

[77] | Clarke, CJS; d’Inverno, RA; Vickers, JA, Combining Cauchy and characteristic codes. I. the vacuum cylindrically symmetric problem, Phys. Rev. D, 52, 6863-6867, (1995) |

[78] | Cook, GB; Huq, MF; Klasky, SA; Scheel, MA; Abrahams, AM; Anderson, A; Anninos, P; Baumgarte, TW; Bishop, NT; Brandt, SR; Browne, JC; Camarda, K; Choptuik, MW; Correll, RR; Evans, CR; Finn, LS; Fox, GC; Gómez, R; Haupt, T; Kidder, LE; Laguna, P; Landry, W; Lehner, L; Lenaghan, J; Marsa, RL; Massó, J; Matzner, RA; Mitra, S; Papadopoulos, P; Parashar, M; Rezzolla, L; Rupright, ME; Saied, F; Saylor, PE; Seidel, E; Shapiro, SL; Shoemaker, DM; Smarr, LL; Suen, W-M; Szilágyi, B; Teukolsky, SA; Putten, MHPM; Walker, P; Winicour, J; York, JW; Binary Black Hole Grand Challenge Alliance, Boosted three-dimensional black-hole evolutions with singularity excision, Phys. Rev. Lett., 80, 2512-2516, (1998) |

[79] | Corkill, RW; Stewart, JM, Numerical relativity. II. numerical methods for the characteristic initial value problem and the evolution of the vacuum field equations for space-times with two Killing vectors, Proc. R. Soc. London, Ser. A, 386, 373-391, (1983) · Zbl 0541.65090 |

[80] | Moerloose, J; Zutter, D, Surface integral representation radiation boundary condition for the FDTD method, IEEE Trans. Ant. Prop., 41, 890-896, (1993) |

[81] | de Oliveira, H.P., and Rodrigues, E.L., “A Galerkin approach for the Bondi problem”, arXiv e-print, (2008). [arXiv:0809.2837]. 3.3.3 |

[82] | Derry, L; Isaacson, RA; Winicour, J, Shear-free gravitational radiation, Phys. Rev., 185, 1647-1655, (1969) |

[83] | d’Inverno, R.A., ed., Approaches to Numerical Relativity, Proceedings of the International Workshop on Numerical Relativity, Southampton, December 1991, (Cambridge University Press, Cambridge; New York, 1992). 2 · Zbl 0772.53003 |

[84] | d’Inverno, RA; Dubal, MR; Sarkies, EA, Cauchy-characteristic matching for a family of cylindrical solutions possessing both gravitational degrees of freedom, Class. Quantum Grav., 17, 3157-3170, (2000) · Zbl 0974.83006 |

[85] | d’Inverno, RA; Vickers, JA, Combining Cauchy and characteristic codes. III. the interface problem in axial symmetry, Phys. Rev. D, 54, 4919-4928, (1996) |

[86] | d’Inverno, RA; Vickers, JA, Combining Cauchy and characteristic codes. IV. the characteristic field equations in axial symmetry, Phys. Rev. D, 56, 772-784, (1997) |

[87] | Dubal, MR; d’Inverno, RA; Clarke, CJS, Combining Cauchy and characteristic codes. II. the interface problem for vacuum cylindrical symmetry, Phys. Rev. D, 52, 6868-6881, (1995) · Zbl 1291.34094 |

[88] | Duff, GFD, Mixed problems for linear systems of first order equations, Can. J. Math., 10, 127-160, (1958) · Zbl 0080.07703 |

[89] | Engquist, B; Majda, A, Absorbing boundary conditions for the numerical simulation of waves, Math. Comput., 31, 629-651, (1977) · Zbl 0367.65051 |

[90] | Fletcher, SJ; Lun, AWC, The Kerr spacetime in generalized Bondi-Sachs coordinates, Class. Quantum Grav., 20, 4153-4167, (2003) · Zbl 1048.83014 |

[91] | Font, J.A., “Numerical Hydrodynamics in General Relativity”, Living Rev. Relativity, 11, lrr-2008-7, (2008). URL (cited on 03 October 2008): http://www.livingreviews.org/lrr-2008-7. 1, 7.1 · Zbl 0944.83007 |

[92] | Frauendiener, J., “Conformal Infinity”, Living Rev. Relativity, 7, lrr-2004-1, (2004). URL (cited on 20 October 2005): http://www.livingreviews.org/lrr-2004-1. 2 |

[93] | Friedman, JL; Schleich, K; Witt, DM, Topological censorship, Phys. Rev. Lett., 71, 1486-1489, (1993) · Zbl 0934.83033 |

[94] | Friedrich, H, The asymptotic characteristic initial value problem for einstein’s vacuum field equations as an initial value problem for a first-order quasilinear symmetric hyperbolic system, Proc. R. Soc. London, Ser. A, 378, 401-421, (1981) · Zbl 0481.58026 |

[95] | Friedrich, H, On the regular and the asymptotic characteristic initial value problem for einstein’s vacuum field equations, Proc. R. Soc. London, Ser. A, 375, 169-184, (1981) · Zbl 0454.58017 |

[96] | Friedrich, H, Cauchy problems for the conformal vacuum field equations in general relativity, Commun. Math. Phys., 91, 445-472, (1983) · Zbl 0555.35116 |

[97] | Friedrich, H, Hyperbolic reductions for einstein’s equations, Class. Quantum Grav., 13, 1451-1469, (1996) · Zbl 0851.58044 |

[98] | Friedrich, H; Nagy, G, The initial boundary value problem for einstein’s vacuum field equation, Commun. Math. Phys., 201, 619-655, (1999) · Zbl 0947.83007 |

[99] | Friedrich, H; Stewart, JM, Characteristic initial data and wavefront singularities in general relativity, Proc. R. Soc. London, Ser. A, 385, 345-371, (1983) · Zbl 0513.58043 |

[100] | Frittelli, S; Gómez, R, Einstein boundary conditions for the 3+1 Einstein equations, Phys. Rev. D, 68, 1-6, (2003) · Zbl 1244.83006 |

[101] | Frittelli, S; Gómez, R, Initial-boundary-value problem of the self-gravitating scalar field in the Bondi-Sachs gauge, Phys. Rev. D, 75, 1-15, (2007) |

[102] | Frittelli, S; Lehner, L, Existence and uniqueness of solutions to characteristic evolution in Bondi-Sachs coordinates in general relativity, Phys. Rev. D, 59, 1-9, (1999) |

[103] | Gallo, E; Lehner, L; Moreschi, OM, Estimating total momentum at finite distances, Phys. Rev. D, 78, 1-11, (2008) |

[104] | Garfinkle, D, Choptuik scaling in null coordinates, Phys. Rev. D, 51, 5558-5561, (1995) |

[105] | Garfinkle, D; Cutler, C; Duncan, GC, Choptuik scaling in six dimensions, Phys. Rev. D, 60, 1-5, (1999) |

[106] | Geroch, R, A method for generating solutions of einstein’s equations, J. Math. Phys., 12, 918-924, (1971) · Zbl 0214.49002 |

[107] | Givoli, D, Non-reflecting boundary conditions, J. Comput. Phys., 94, 1-29, (1991) · Zbl 0731.65109 |

[108] | Gnedin, ML; Gnedin, NY, Destruction of the Cauchy horizon in the Reissner-Nordström black hole, Class. Quantum Grav., 10, 1083-1102, (1993) |

[109] | Goldwirth, DS; Piran, T, Gravitational collapse of massless scalar field and cosmic censorship, Phys. Rev. D, 36, 3575-3581, (1987) |

[110] | Gómez, R, Gravitational waveforms with controlled accuracy, Phys. Rev. D, 64, 1-8, (2001) |

[111] | Gómez, R; Barreto, W; Frittelli, S, Framework for large-scale relativistic simulations in the characteristic approach, Phys. Rev. D, 76, 1-22, (2007) |

[112] | Gómez, R; Frittelli, S, First-order quasilinear canonical representation of the characteristic formulation of the Einstein equations, Phys. Rev. D, 68, 1-6, (2003) |

[113] | Gómez, R; Husa, S; Lehner, L; Winicour, J, Gravitational waves from a fissioning white hole, Phys. Rev. D, 66, 1-8, (2002) |

[114] | Gómez, R; Husa, S; Winicour, J, Complete null data for a black hole collision, Phys. Rev. D, 64, 1-20, (2001) |

[115] | Gómez, R; Laguna, P; Papadopoulos, P; Winicour, J, Cauchy-characteristic evolution of Einstein-Klein-Gordon systems, Phys. Rev. D, 54, 4719-4727, (1996) |

[116] | Gómez, R; Lehner, L; Marsa, RL; Winicour, J, Moving black holes in 3D, Phys. Rev. D, 57, 4778-4788, (1998) |

[117] | Gómez, R; Lehner, L; Marsa, RL; Winicour, J; Abrahams, AM; Anderson, A; Anninos, P; Baumgarte, TW; Bishop, NT; Brandt, SR; Browne, JC; Camarda, K; Choptuik, MW; Cook, GB; Correll, RR; Evans, CR; Finn, LS; Fox, GC; Haupt, T; Huq, MF; Kidder, LE; Klasky, SA; Laguna, P; Landry, W; Lenaghan, J; Massó, J; Matzner, RA; Mitra, S; Papadopoulos, P; Parashar, M; Rezzolla, L; Rupright, ME; Saied, F; Saylor, PE; Scheel, MA; Seidel, E; Shapiro, SL; Shoemaker, D; Smarr, LL; Szilágyi, B; Teukolsky, SA; Putten, MHPM; Walker, P; York, JW; The Binary Black Hole Grand Challenge Alliance, Stable characteristic evolution of generic three-dimensional single-black-hole spacetimes, Phys. Rev. Lett., 80, 3915-3918, (1998) |

[118] | Gómez, R; Lehner, L; Papadopoulos, P; Winicour, J, The eth formalism in numerical relativity, Class. Quantum Grav., 14, 977-990, (1997) · Zbl 0872.53054 |

[119] | Gómez, R; Marsa, RL; Winicour, J, Black hole excision with matching, Phys. Rev. D, 56, 6310-6319, (1997) |

[120] | Gómez, R; Papadopoulos, P; Winicour, J, Null cone evolution of axisymmetric vacuum space-times, J. Math. Phys., 35, 4184-4204, (1994) · Zbl 0814.35136 |

[121] | Gómez, R; Reilly, P; Winicour, J; Isaacson, RA, Post-Newtonian behavior of the Bondi mass, Phys. Rev. D, 47, 3292-3302, (1993) |

[122] | Gómez, R; Winicour, J, Asymptotics of gravitational collapse of scalar waves, J. Math. Phys., 33, 1445-1457, (1992) |

[123] | Gómez, R; Winicour, J, Gravitational wave forms at finite distances and at null infinity, Phys. Rev. D, 45, 2776-2782, (1992) · Zbl 1232.83033 |

[124] | Gómez, R; Winicour, J; Isaacson, RA, Evolution of scalar fields from characteristic data, J. Comput. Phys., 98, 11-25, (1992) · Zbl 0747.65080 |

[125] | Gómez, R; Winicour, J; Schmidt, BG, Newman-Penrose constants and the tails of self-gravitating waves, Phys. Rev. D, 49, 2828-2836, (1994) |

[126] | Grote, MJ; Keller, JB, Nonreflecting boundary conditions for maxwell’s equations, J. Comput. Phys., 139, 327-342, (1998) · Zbl 0908.65118 |

[127] | Gundlach, C., and Martín-García, J.M., “Critical Phenomena in Gravitational Collapse”, Living Rev. Relativity, 10, lrr-2007-5, (1999). URL (cited on 03 October 2008): http://www.livingreviews.org/lrr-2007-5. 3.1 · Zbl 0973.83004 |

[128] | Gundlach, C; Martín-García, JM, Symmetric hyperbolicity and consistent boundary conditions for second-order Einstein equations, Phys. Rev. D, 70, 1-16, (2004) |

[129] | Gundlach, C; Price, RH; Pullin, J, Late-time behavior of stellar collapse and explosions. I. linearized perturbations, Phys. Rev. D, 49, 883-889, (1994) |

[130] | Gundlach, C; Price, RH; Pullin, J, Late-time behavior of stellar collapse and explosions. II. nonlinear evolution, Phys. Rev. D, 49, 890-899, (1994) |

[131] | Gustafsson, B; Kreiss, H-O, Boundary conditions for time dependent problems with an artificial boundary, J. Comput. Phys., 30, 331-351, (1979) · Zbl 0431.65062 |

[132] | Gustafsson, B., Kreiss, H.-O., and Oliger, J., Time Dependent Problems and Difference Methods, (Wiley, New York, 1995). 5.1 · Zbl 0843.65061 |

[133] | Hagstrom, T; Hariharan, SI, Accurate boundary conditions for exterior problems in gas dynamics, Math. Comput., 51, 581-597, (1988) · Zbl 0699.76083 |

[134] | Hamadé, RS; Horne, JH; Stewart, JM, Continuous self-similarity and \(S\)-duality, Class. Quantum Grav., 13, 2241-2253, (1996) · Zbl 0858.53075 |

[135] | Hamadé, RS; Stewart, JM, The spherically symmetric collapse of a massless scalar field, Class. Quantum Grav., 13, 497-512, (1996) · Zbl 0849.53068 |

[136] | Hayward, SA, Dual-null dynamics of the Einstein field, Class. Quantum Grav., 10, 779-790, (1993) · Zbl 0773.53047 |

[137] | Hedstrom, GW, Nonreflecting boundary conditions for nonlinear hyperbolic systems, J. Comput. Phys., 30, 222-237, (1979) · Zbl 0397.35043 |

[138] | Higdon, RL, Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation, Math. Comput., 47, 437-459, (1986) · Zbl 0609.35052 |

[139] | Hod, S, High-order contamination in the tail gravitational collapse, Phys. Rev. D, 60, 1-4, (1999) |

[140] | Hod, S, Wave tails in non-trivial backgrounds, Class. Quantum Grav., 18, 1311-1318, (2001) · Zbl 0978.83021 |

[141] | Hod, S, Wave tails in time-dependent backgrounds, Phys. Rev. D, 66, 1-4, (2002) |

[142] | Hod, S; Piran, T, Critical behavior and universality in gravitational collapse of a charged scalar field, Phys. Rev. D, 55, 3485-3496, (1997) |

[143] | Hod, S; Piran, T, Late-time evolution of charged gravitational collapse and decay of charged scalar hair. I, Phys. Rev. D, 58, 1-6, (1998) |

[144] | Hod, S; Piran, T, Late-time evolution of charged gravitational collapse and decay of charged scalar hair. II, Phys. Rev. D, 58, 1-6, (1998) |

[145] | Hod, S; Piran, T, Late-time evolution of charged gravitational collapse and decay of charged scalar hair. III. nonlinear analysis, Phys. Rev. D, 58, 1-6, (1998) |

[146] | Hod, S; Piran, T, Late-time tails in gravitational collapse of a self-interacting (massive) scalar-field and decay of a self-interacting scalar hair, Phys. Rev. D, 58, 1-6, (1998) |

[147] | Hod, S; Piran, T, Mass inflation in dynamical gravitational collapse of a charged scalar field, Phys. Rev. Lett., 81, 1554-1557, (1998) · Zbl 0962.83026 |

[148] | Husa, S; Fernádez-Jambrina, L (ed.); González-Romero, LM (ed.), Numerical relativity with the conformal field equations, Proceedings of the 24th Spanish Relativity Meeting on Relativistic Astrophysics, Madrid, 2001, Berlin; New York |

[149] | Husa, S; Lechner, C; Pürrer, M; Thornburg, J; Aichelburg, PC, Type II critical collapse of a self-gravitating nonlinear \(σ\) model, Phys. Rev. D, 62, 1-11, (2000) |

[150] | Husa, S; Winicour, J, Asymmetric merger of black holes, Phys. Rev. D, 60, 1-13, (1999) |

[151] | Husa, S; Zlochower, Y; Gómez, R; Winicour, J, Retarded radiation from colliding black holes in the close limit, Phys. Rev. D, 65, 1-14, (2002) |

[152] | Ipser, JR; Horwitz, G, The problem of maximizing functionals in Newtonian stellar dynamics, and its relation to thermodynamic and dynamical stability, Astrophys. J., 232, 863-873, (1979) |

[153] | Isaacson, RA; Welling, JS; Winicour, J, Null cone computation of gravitational radiation, J. Math. Phys., 24, 1824-1834, (1983) |

[154] | Israeli, M; Orszag, SA, Approximation of radiation boundary conditions, J. Comput. Phys., 41, 115-135, (1981) · Zbl 0469.65082 |

[155] | Jiang, H; Wong, YS, Absorbing boundary conditions for second-order hyperbolic equations, J. Comput. Phys., 88, 205-231, (1990) · Zbl 0701.65086 |

[156] | Kates, RE; Kegeles, LS, Nonanalytic terms in the slow-motion expansion of a radiating scalar field on a Schwarzschild background, Phys. Rev. D, 25, 2030-2037, (1982) |

[157] | Khan, KA; Penrose, R, Scattering of two impulsive gravitational plane waves, Nature, 229, 185-186, (1971) |

[158] | Kreiss, H.-O., and Oliger, J., Methods for the approximate solution of time dependent problems, GARP Publications Series, 10, (World Meteorological Organization (WMO), International Council of Scientific Unions (ICSU), Geneva, 1973). Global Atmospheric Research Programme. 4.2.1 |

[159] | Kreiss, H-O; Ortiz, OE; Frauendiener, J (ed.); Friedrich, H (ed.), Some mathematical and numerical questions connected with first and second order time-dependent systems of partial differential equations, Proceedings of the international workshop, Tübingen, Germany, 2-4 April 2001, Berlin; New York · Zbl 1043.35525 |

[160] | Kreiss, H-O; Reula, O; Sarbach, O; Winicour, J, Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates, Class. Quantum Grav., 24, 5973-5984, (2007) · Zbl 1130.83004 |

[161] | Kreiss, H.-O., Reula, O., Sarbach, O., and Winicour, J., “Boundary conditions for coupled quasilinear wave equations with application to isolated systems”, arXiv e-print, (2008). [arXiv:0807.3207]. 5.3 · Zbl 1172.35077 |

[162] | Kreiss, H-O; Winicour, J, Problems which are well posed in a generalized sense with applications to the Einstein equations, Class. Quantum Grav., 23, s405-s420, (2006) · Zbl 1191.83010 |

[163] | Lehner, L, A dissipative algorithm for wave-like equations in the characteristic formulation, J. Comput. Phys., 149, 59-74, (1999) · Zbl 0923.65058 |

[164] | Lehner, L, Matching characteristic codes: exploiting two directions, Int. J. Mod. Phys. D, 9, 459-473, (2000) · Zbl 0974.83001 |

[165] | Lehner, L; Bishop, NT; Gómez, R; Szilágyi, B; Winicour, J, Exact solutions for the intrinsic geometry of black hole coalescence, Phys. Rev. D, 60, 1-10, (1999) |

[166] | Lehner, L., Gómez, R., Husa, S., Szilágyi, B., Bishop, N.T., and Winicour, J., “Bagels Form When Black Holes Collide”, institutional homepage, Pittsburgh Supercomputing Center. URL (cited on 30 July 2005): http://www.psc.edu/research/graphics/gallery/winicour.html. 4.3 |

[167] | Lehner, L; Moreschi, OM, Dealing with delicate issues in waveform calculations, Phys. Rev. D, 76, 1-12, (2007) |

[168] | Lindman, EL, ‘free-space’ boundary conditions for the time dependent wave equation, J. Comput. Phys., 18, 66-78, (1975) · Zbl 0417.73042 |

[169] | Linke, F; Font, JA; Janka, H-T; Müller, E; Papadopoulos, P, Spherical collapse of supermassive stars: neutrino emission and gamma-ray bursts, Astron. Astrophys., 376, 568-579, (2001) |

[170] | Lousto, CO; Price, RH, Understanding initial data for black hole collisions, Phys. Rev. D, 56, 6439-6457, (1997) |

[171] | Marsa, RL; Choptuik, MW, Black-hole-scalar-field interactions in spherical symmetry, Phys. Rev. D, 54, 4929-4943, (1996) |

[172] | Matzner, RA; Seidel, E; Shapiro, SL; Smarr, LL; Suen, W-M; Teukolsky, SA; Winicour, J, Geometry of a black hole collision, Science, 270, 941-947, (1995) |

[173] | May, MM; White, RH, Hydrodynamic calculations of general-relativistic collapse, Phys. Rev., 141, 1232-1241, (1966) |

[174] | Miller, JC; Motta, S, Computations of spherical gravitational collapse using null slicing, Class. Quantum Grav., 6, 185-193, (1989) |

[175] | Moncrief, V, Gravitational perturbations of spherically symmetric systems. I the exterior problem, Ann. Phys. (N.Y.), 88, 323-342, (1974) |

[176] | Müller Zum Hagen, H; Seifert, H-J, On characteristic initial-value and mixed problems, Gen. Relativ. Gravit., 8, 259-301, (1977) · Zbl 0417.35052 |

[177] | Nagar, A; Rezzolla, L, Gauge-invariant non-spherical metric perturbations of Schwarzschild black-hole spacetime, Class. Quantum Grav., 22, r167-r192, (2005) · Zbl 1078.83024 |

[178] | Nayfeh, A., Perturbation Methods, (Wiley, New York, 1973). 5.4 · Zbl 0265.35002 |

[179] | Newman, ET; Penrose, R, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys., 3, 566-578, (1962) · Zbl 0108.40905 |

[180] | Newman, ET; Penrose, R, Note on the Bondi-metzner-Sachs group, J. Math. Phys., 7, 863, (1966) |

[181] | Newman, ET; Penrose, R, New conservation laws for zero rest-mass fields in asymptotically flat space-time, Proc. R. Soc. London, Ser. A, 305, 175-204, (1968) |

[182] | Oren, Y; Piran, T, Collapse of charged scalar fields, Phys. Rev. D, 68, 1-12, (2003) · Zbl 1244.83023 |

[183] | Papadopoulos, P, Nonlinear harmonic generation in finite amplitude black hole oscillations, Phys. Rev. D, 65, 1-11, (2002) |

[184] | Papadopoulos, P; Font, JA, Relativistic hydrodynamics on spacelike and null surfaces: formalism and computations of spherically symmetric spacetimes, Phys. Rev. D, 61, 1-15, (2000) |

[185] | Papadopoulos, P; Font, JA, Imprints of accretion on gravitational waves from black holes, Phys. Rev. D, 63, 1-5, (2001) |

[186] | Papadopoulos, P.O., Algorithms for the gravitational characteristic initial value problem, Ph.D. Thesis, (University of Pittsburgh, Pittsburgh, 1994). [ADS]. 3.3, 7.2 |

[187] | Penrose, R, Asymptotic properties of fields and space-times, Phys. Rev. Lett., 10, 66-68, (1963) |

[188] | Penrose, R, Gravitational collapse: the role of general relativity, Riv. Nuovo Cimento, 1, 252-276, (1969) |

[189] | Phillips, NA; Syono, S (ed.), A map projection system suitable for large-scale numerical weather prediction, 262-267, (1957), Tokyo |

[190] | Piran, T, Numerical codes for cylindrical general relativistic systems, J. Comput. Phys., 35, 254-283, (1980) · Zbl 0424.65062 |

[191] | Piran, T; Safier, PN; Katz, J, Cylindrical gravitational waves with two degrees of freedom: an exact solution, Phys. Rev. D, 34, 331-332, (1986) |

[192] | Piran, T; Safier, PN; Stark, RF, General numerical solution of cylindrical gravitational waves, Phys. Rev. D, 32, 3101-3107, (1985) |

[193] | Poisson, E; Israel, W, Internal structure of black holes, Phys. Rev. D, 41, 1796-1809, (1990) |

[194] | Pollney, D., Algebraic and numerical techniques in general relativity, Ph.D. Thesis, (University of Southampton, Southampton, 2000). 3.3.4, 5.6 |

[195] | Pretorius, F, Evolution of binary black-hole spacetimes, Phys. Rev. Lett., 95, 121101, (2005) |

[196] | Pretorius, F; Israel, W, Quasi-spherical light cones of the Kerr geometry, Class. Quantum Grav., 15, 2289-2301, (1998) · Zbl 0969.83009 |

[197] | Pretorius, F; Lehner, L, Adaptive mesh refinement for characteristic codes, J. Comput. Phys., 198, 10-34, (2004) · Zbl 1052.65090 |

[198] | Price, RH, Nonspherical perturbations of relativistic gravitational collapse. I. scalar and gravitational perturbations, Phys. Rev. D, 5, 2419-2438, (1972) |

[199] | Price, RH; Pullin, J, Colliding black holes: the close limit, Phys. Rev. Lett., 72, 3297-3300, (1994) · Zbl 0973.83532 |

[200] | Regge, T; Wheeler, JA, Stability of a Schwarzschild singularity, Phys. Rev., 108, 1063-1069, (1957) · Zbl 0079.41902 |

[201] | Reisswig, C; Bishop, NT; Lai, CW; Thornburg, J; Szilágyi, B, Characteristic evolutions in numerical relativity using six angular patches, Class. Quantum Grav., 24, s237-s339, (2007) · Zbl 1117.83013 |

[202] | Renaut, RA, Absorbing boundary conditions, difference operators, and stability, J. Comput. Phys., 102, 236-251, (1992) · Zbl 0766.65070 |

[203] | Rendall, A.D., “Local and Global Existence Theorems for the Einstein Equations”, Living Rev. Relativity, 8, lrr-2005-6, (2000). URL (cited on 03 October 2008): http://www.livingreviews.org/lrr-2005-6. 2 |

[204] | Rezzolla, L; Abrahams, AM; Matzner, RA; Rupright, ME; Shapiro, SL, Cauchy-perturbative matching and outer boundary conditions: computational studies, Phys. Rev. D, 59, 1-17, (1999) |

[205] | Rinne, O; Lindblom, L; Scheel, MA, Testing outer boundary treatments for the Einstein equations, Class. Quantum Grav., 24, 4053-4078, (2007) · Zbl 1205.83009 |

[206] | Ronchi, C; Iacono, R; Paolucci, PS, The ‘cubed sphere’: A new method for the solution of partial differential equations in spherical geometry, J. Comput. Phys., 124, 93-114, (1996) · Zbl 0849.76049 |

[207] | Ruiz, M; Rinne, O; Sarbach, O, Outer boundary conditions for einstein’s field equations in harmonic coordinates, Class. Quantum Grav., 24, 6349-6377, (2007) · Zbl 1197.83020 |

[208] | Rupright, ME; Abrahams, AM; Rezzolla, L, Cauchy-perturbative matching and outer boundary conditions: methods and tests, Phys. Rev. D, 58, 1-9, (1998) |

[209] | Ryaben’kii, V; Tsynkov, SV; Hafez, M (ed.); Oshima, K (ed.), An application of the difference potentials method to solving external problems in CFD, (1998), Singapore; River Edge |

[210] | Sachs, RK, Asymptotic symmetries in gravitational theory, Phys. Rev., 128, 2851-2864, (1962) · Zbl 0114.21202 |

[211] | Sachs, RK, Gravitational waves in general relativity. VIII. waves in asymptotically flat space-time, Proc. R. Soc. London, Ser. A, 270, 103-126, (1962) · Zbl 0101.43605 |

[212] | Sachs, RK, On the characteristic initial value problem in gravitational theory, J. Math. Phys., 3, 908-914, (1962) · Zbl 0111.42202 |

[213] | Sarbach, O, Absorbing boundary conditions for einstein’s field equations, J. Phys.: Conf. Ser., 91, 012005, (2007) |

[214] | Scheel, MA; Shapiro, SL; Teukolsky, SA, Collapse to black holes in Brans-Dicke theory. I. horizon boundary conditions for dynamical spacetimes, Phys. Rev. D, 51, 4208-4235, (1995) |

[215] | Scheel, MA; Shapiro, SL; Teukolsky, SA, Collapse to black holes in Brans-Dicke theory. II. comparison with general relativity, Phys. Rev. D, 51, 4236-4249, (1995) |

[216] | Seidel, E; Suen, W-M, Dynamical evolution of boson stars: perturbing the ground state, Phys. Rev. D, 42, 384-403, (1990) |

[217] | Shapiro, SL; Teukolsky, SA; Winicour, J, Toroidal black holes and topological censorship, Phys. Rev. D, 52, 6982-6987, (1995) |

[218] | Shibata, M; Nakamura, T, Evolution of three-dimensional gravitational waves: harmonic slicing case, Phys. Rev. D, 52, 5428-5444, (1995) · Zbl 1250.83027 |

[219] | Siebel, F., Simulation of axisymmetric flows in the characteristic formulation of general relativity, Ph.D. Thesis, (Technische Universität München, München, 2002). Related online version (cited on 14 April 2009): http://tumb1.biblio.tu-muenchen.de/publ/diss/ph/2002/siebel.html. 1,7.2 |

[220] | Siebel, F; Font, JA; Müller, E; Papadopoulos, P, Simulating the dynamics of relativistic stars via a light-cone approach, Phys. Rev. D, 65, 1-15, (2002) |

[221] | Siebel, F; Font, JA; Müller, E; Papadopoulos, P, Axisymmetric core collapse simulations using characteristic numerical relativity, Phys. Rev. D, 67, 1-16, (2003) |

[222] | Siebel, F; Font, JA; Papadopoulos, P, Scalar field induced oscillations of relativistic stars and gravitational collapse, Phys. Rev. D, 65, 1-10, (2001) |

[223] | Sjödin, KRP; Sperhake, U; Vickers, JA, Dynamic cosmic strings. I, Phys. Rev. D, 63, 1-14, (2001) |

[224] | Sod, G.A., Numerical Methods in Fluid Dynamics: Initial and Initial Boundary-Value Problems, (Cambridge University Press, Cambridge; New York, 1985). 5.1 · Zbl 0592.76001 |

[225] | Sorkin, E; Piran, T, Effects of pair creation on charged gravitational collapse, Phys. Rev. D, 63, 1-12, (2001) |

[226] | Sorkin, RD, A criterion for the onset of instability at a turning point, Astrophys. J., 249, 254-257, (1981) |

[227] | Sperhake, U; Sjödin, KRP; Vickers, JA, Dynamic cosmic strings. II. numerical evolution of excited strings, Phys. Rev. D, 63, 1-15, (2001) |

[228] | Stark, RF; Piran, T, A general relativistic code for rotating axisymmetric configurations and gravitational radiation: numerical methods and tests, Comput. Phys. Rep., 5, 221-264, (1987) |

[229] | Stewart, JM; Bonnor, WB (ed.); Islam, JN (ed.); MacCallum, MAH (ed.), Numerical relativity, Proceedings of the Conference on Classical (Non-Quantum) General Relativity, City University, London, 21-22 December 1983, Cambridge; New York |

[230] | Stewart, JM; Winkler, K-HA (ed.); Norman, ML (ed.), The characteristic initial value problem in general relativity, Proceedings of the NATO Advanced Research Workshop on Astrophysical Radiation Hydrodynamics, Garching, Germany, August 2-13, 1982, Dordrecht; Boston |

[231] | Stewart, JM, Numerical relativity III. the Bondi mass revisited, Proc. R. Soc. London, Ser. A, 424, 211-222, (1989) · Zbl 0683.65112 |

[232] | Stewart, JM, The Cauchy problem and the initial boundary value problem in numerical relativity, Class. Quantum Grav., 15, 2865-2889, (1998) · Zbl 0941.83004 |

[233] | Stewart, JM; Friedrich, H, Numerical relativity. I. the characteristic initial value problem, Proc. R. Soc. London, Ser. A, 384, 427-454, (1982) · Zbl 0541.65089 |

[234] | Szilágyi, B., Cauchy-characteristic matching in general relativity, Ph.D. Thesis, (University of Pittsburgh, Pittsburgh, 2000). [ADS]. 5 |

[235] | Szilágyi, B; Gómez, R; Bishop, NT; Winicour, J, Cauchy boundaries in linearized gravitational theory, Phys. Rev. D, 62, 1-10, (2000) |

[236] | Szilágyi, B; Winicour, J, Well-posed initial-boundary evolution in general relativity, Phys. Rev. D, 68, 1-5, (2003) · Zbl 1244.83010 |

[237] | Tamburino, LA; Winicour, J, Gravitational fields in finite and conformal Bondi frames, Phys. Rev., 150, 1039-1053, (1966) |

[238] | Teukolsky, SA, Perturbations of a rotating black hole. I. fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations, Astrophys. J., 185, 635-647, (1973) |

[239] | Teukolsky, SA, Linearized quadrupole waves in general relativity and the motion of test particles, Phys. Rev. D, 26, 745-750, (1982) |

[240] | Thompson, KW, Time dependent boundary conditions for hyperbolic systems, J. Comput. Phys., 68, 1-24, (1987) · Zbl 0619.76089 |

[241] | Thornburg, J, Black-hole excision with multiple grid patches, Class. Quantum Grav., 21, 3665-3691, (2004) · Zbl 1068.83012 |

[242] | Thornburg, J, A fast apparent horizon finder for three-dimensional Cartesian grids in numerical relativity, Class. Quantum Grav., 21, 743-766, (2004) · Zbl 1045.83006 |

[243] | Ting, L; Miksis, MJ, Exact boundary conditions for scattering problems, J. Acoust. Soc. Am., 80, 1825-1827, (1986) |

[244] | Trefethen, LN; Halpern, L, Well-posedness of one-way wave equations and absorbing boundary conditions, Math. Comput., 47, 421-435, (1986) · Zbl 0618.65077 |

[245] | Tsynkov, S.V., Artificial Boundary Conditions Based on the Difference Potentials Method, NASA Technical Memorandum, 110265, (NASA Langley Research Center, Hampton, 1996). Related online version (cited on 04 February 2009): http://hdl.handle.net/2060/19960045440. 5.1 · Zbl 1063.76622 |

[246] | Wald, R.M., General Relativity, (University of Chicago Press, Chicago, 1984). 5.5.3 · Zbl 0549.53001 |

[247] | Weber, J; Wheeler, JA, Reality of the cylindrical gravitational waves of Einstein and Rosen, Rev. Mod. Phys., 29, 509-515, (1957) · Zbl 0078.43204 |

[248] | Winicour, J, Newtonian gravity on the null cone, J. Math. Phys., 24, 1193-1198, (1983) · Zbl 0522.76128 |

[249] | Winicour, J, Null infinity from a quasi-Newtonian view, J. Math. Phys., 25, 2506-2514, (1984) |

[250] | Winicour, J, The quadrupole radiation formula, Gen. Relativ. Gravit., 19, 281-287, (1987) |

[251] | Winicour, J, The characteristic treatment of black holes, Prog. Theor. Phys. Suppl., 136, 57-71, (1999) · Zbl 0986.83021 |

[252] | Xanthopoulos, BC, Cylindrical waves and cosmic strings of Petrov type D, Phys. Rev. D, 34, 3608-3616, (1986) |

[253] | York, JW; Smarr, LL (ed.), Kinematics and dynamics of general relativity, Proceedings of the Battelle Seattle Workshop, July 24-August 4, 1978, Cambridge; New York |

[254] | Zerilli, FJ, Gravitational field of a particle falling in a Schwarzschild geometry analyzed in tensor harmonics, Phys. Rev. D, 2, 2141-2160, (1970) · Zbl 1227.83025 |

[255] | Zlochower, Y., Waveforms from colliding black holes, Ph.D. Thesis, (University of Pittsburgh, Pittsburgh, 2002). [ADS]. 1, 4.2.5, 4.4, 4.5, 4.5, 6 |

[256] | Zlochower, Y; Gómez, R; Husa, S; Lehner, L; Winicour, J, Mode coupling in the nonlinear response of black holes, Phys. Rev. D, 68, 1-16, (2003) |

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.