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General relativistic null-cone evolutions with a high-order scheme. (English) Zbl 1269.83014
Summary: We present a high-order scheme for solving the full non-linear Einstein equations on characteristic null hypersurfaces using the framework established by Bondi and Sachs. This formalism allows asymptotically flat spaces to be represented on a finite, compactified grid, and is thus ideal for far-field studies of gravitational radiation. We have designed an algorithm based on 4th-order radial integration and finite differencing, and a spectral representation of angular components. Consequently the scheme offers more accuracy at a given computational cost compared to previous methods which are second-order accurate. Based on a newly implemented code, we show that the new numerical scheme remains stable and is convergent at the expected order of accuracy.

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C57 Black holes
83-08 Computational methods for problems pertaining to relativity and gravitational theory
85A25 Radiative transfer in astronomy and astrophysics
83C35 Gravitational waves
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
85A04 General questions in astronomy and astrophysics
Full Text: DOI
[1] Reisswig, C; Bishop, NT; Pollney, D; Szilagyi, B, Unambiguous determination of gravitational waveforms from binary black hole mergers, Phys. Rev. Lett., 103, 221101, (2009)
[2] Reisswig, C; Bishop, NT; Pollney, D; Szilagyi, B, Characteristic extraction in numerical relativity: binary black hole merger waveforms at null infinity, Class. Quantum Gravity, 27, 075014, (2010) · Zbl 1187.83026
[3] Babiuc, MC; Szilagyi, B; Winicour, J; Zlochower, Y, A characteristic extraction tool for gravitational waveforms, Phys. Rev. D, 84, 044057, (2011)
[4] Babiuc, MC; Winicour, J; Zlochower, Y, Binary black hole waveform extraction at null infinity, Class. Quantum Gravity, 28, 134006, (2011) · Zbl 1219.83032
[5] Reisswig, C; Ott, CD; Sperhake, U; Schnetter, E, Gravitational wave extraction in simulations of rotating stellar core collapse, Phys. Rev. D, 83, 064008, (2011)
[6] Ott, CD; Reisswig, C; Schnetter, E; O’Connor, E; Sperhake, U; Löffler, F; Diener, P; Abdikamalov, E; Hawke, I; Burrows, A, Dynamics and gravitational wave signature of collapsar formation, Phys. Rev. Lett., 106, 161103, (2011)
[7] Winicour, J.: Characteristic evolution and matching. Living Rev. Relativ. 8, 10 (2005) [Online article]. · Zbl 1316.83016
[8] Bondi, H; Burg, MGJ; Metzner, AWK, Gravitational waves in general relativity VII. waves from axi-symmetric isolated systems, Proc. R. Soc. A., 269, 21-52, (1962) · Zbl 0106.41903
[9] Sachs, RK, Gravitational waves in general relativity, Proc. R. Soc. A., 270, 103-126, (1962) · Zbl 0101.43605
[10] Penrose, R, Asymptotic properties of fields and spacetimes, Phys. Rev. Lett., 10, 66-68, (1963)
[11] Bishop, NT; Gómez, R; Lehner, L; Winicour, J, Cauchy-characteristic extraction in numerical relativity, Phys. Rev. D., 54, 6153-6165, (1996)
[12] Bishop, NT; Gómez, R; Lehner, L; Maharaj, M; Winicour, J, High-powered gravitational news, Phys. Rev. D., 56, 6298-6309, (1997)
[13] Löffler, F; Faber, J; Bentivegna, E; Bode, T; Diener, P; Haas, R; Hinder, I; Mundim, BC; Ott, CD; Schnetter, E; Allen, G; Campanelli, M; Laguna, P, The Einstein toolkit: a community computational infrastructure for relativistic astrophysics, Class. Quantum Gravity, 29, 115001, (2012) · Zbl 1247.83003
[14] Gómez, R; Winicour, J; Isaacson, R, Evolution of scalar fields from characteristic data, J. Comput. Phys., 98, 11-25, (1992) · Zbl 0747.65080
[15] Reisswig, C; Bishop, NT; Lai, CW; Thornburg, J; Szilágyi, B, Characteristic evolutions in numerical relativity using six angular patches, Class. Quantum Gravity, 24, s327-s339, (2007) · Zbl 1117.83013
[16] Gomez, R., Barreto, W., Frittelli, S.: A framework for large-scale relativistic simulations in the characteristic approach. Phys. Rev. D 76, 124029 (2007) · Zbl 0101.43605
[17] Bartnik, R, Einstein equations in the null quasispherical gauge, Class. Quantum Gravity, 14, 2185-2194, (1997) · Zbl 0877.53059
[18] Bartnik, R.A., Norton, A.H.: Einstein equations in the null quasi-spherical gauge III: numerical algorithms. gr-qc/9904045 (1999)
[19] Bishop, NT; Clarke, C; d’Inverno, R, Numerical relativity on a transputer array, Class. Quantum Gravity, 7, l23-l27, (1990) · Zbl 1051.78001
[20] Stewart, J.M.: Advanced General Relativity. Cambridge University Press, Cambridge (1990) · Zbl 0752.53048
[21] Gustafsson, B., Gustafsson, H.O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, New York (1995) · Zbl 0843.65061
[22] Gómez, R; Lehner, L; Papadopoulos, P; Winicour, J, The eth formalism in numerical relativity, Class. Quantum Gravity, 14, 977-990, (1997) · Zbl 0872.53054
[23] Goldberg, JN; MacFarlane, AJ; Newman, ET; Rohrlich, F; Sudarshan, ECG, Spin-\(s\) spherical harmonics and \(ð \), J. Math. Phys., 8, 2155-2161, (1967) · Zbl 0155.57402
[24] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C++ : The Art of Scientific Computing. New York, 3rd edition (2002) · Zbl 1078.65500
[25] Driscoll, JR; Healy, DM, Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250, (1994) · Zbl 0801.65141
[26] Goodale, T., Allen, G., Lanfermann, G., Massó, J., Radke, T., Seidel, E., Shalf, J.: The Cactus framework and toolkit: design and applications. In: Vector and Parallel Processing—VECPAR’2002, 5th International Conference, Lecture Notes in Computer Science, Berlin, Springer (2003) · Zbl 1027.65524
[27] http://www.cactuscode.org · Zbl 0872.53054
[28] Schnetter, E; Hawley, SH; Hawke, I, Evolutions in 3D numerical relativity using fixed mesh refinement, Class. Quantum Gravity, 21, 1465-1488, (2004) · Zbl 1047.83002
[29] http://www.carpetcode.org
[30] Reisswig, C.: Binary Black Hole Mergers and Novel Approaches to Gravitational Wave Extraction in Numerical Relativity. PhD thesis, Leibniz Universität Hannover, (2010)
[31] Pollney, D., Reisswig, C., Schnetter, E., Dorband, N., Diener, P.: High accuracy binary black hole simulations with an extended wave zone. arXiv:0910.3803, (2009) · Zbl 0747.65080
[32] Pollney, D; Reisswig, C; Dorband, N; Schnetter, E; Diener, P, The asymptotic falloff of local waveform measurements in numerical relativity, Phys. Rev. D, 80, 121502, (2009)
[33] 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
[34] Bishop, NT, Linearized solutions of the Einstein equations within a Bondi-Sachs framework, and implications for boundary conditions in numerical simulations, Class. Quantum Gravity, 22, 2393-2406, (2005) · Zbl 1077.83013
[35] Szilágyi, B; Gomez, R; Bishop, NT; Winicour, J, Cauchy boundaries in linearized gravitational theory, Phys. Rev. D., 62, 104006, (2000)
[36] Alcubierre, M; Allen, G; Baumgarte, TW; Bona, C; Fiske, D; Goodale, T; Guzmán, FS; Hawke, I; Hawley, S; Husa, S; Koppitz, M; Lechner, C; Lindblom, L; Pollney, D; Rideout, D; Salgado, M; Schnetter, E; Seidel, E; Shinkai, H; Shoemaker, D; Szilágyi, B; Takahashi, R; Winicour, J, Towards standard testbeds for numerical relativity, Class. Quantum Grav., 21, 589-613, (2004) · Zbl 1045.83005
[37] Diener, P; Dorband, EN; Schnetter, E; Tiglio, M, Optimized high-order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions, J. Sci. Comput., 32, 109-145, (2007) · Zbl 1120.65092
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