Multidomain spectral method for the helically reduced wave equation

Journal of Computational Physics

Volumn 227, Issue 2, 2007, Pages 1126-1161

  Lau, Stephen R ^a,b   Price, Richard H ^c  

^a UNIVERSITY OF NEW MEXICO   (United States)

^b BROWN UNIVERSITY   (United States)

^c UNIVERSITY OF TEXAS AT BROWNSVILLE   (United States)

Author keywords

35L05; 35M10; 5L20; 65M70; 83C35; Gravitational waves; Helical symmetry; Mixed PDE; Spectral methods

Indexed keywords

DOMAIN DECOMPOSITION METHODS; GRAVITATIONAL EFFECTS; GRAVITY WAVES; NUMERICAL METHODS;

35L05; 35M10; 5L20; 65M70; 83C35; GRAVITATIONAL-WAVES; HELICAL SYMMETRY; MIXED PDE; SPECTRAL METHODS; STANDING WAVE APPROXIMATION;

WAVE EQUATIONS;

EID: 36148977247     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.08.032     Document Type: Article

References (55)

1. Pretorius F. Evolution of binary black-hole spacetimes. Phys. Rev. Lett. 95 (2005) 121101
2. Campanelli M., Lousto C.O., Marronetti P., and Zlochower Y. Accurate evolutions of orbiting black-hole binaries without excision. Phys. Rev. Lett. 96 (2006) 111101
3. Campanelli M., Lousto C.O., and Zlochower Y. Spinning-black-hole binaries: the orbital hang-up. Phys. Rev. D 74 (2006) 041501
4. M. Campanelli, C.O. Lousto, Y. Zlochower, B. Krishnan, D. Merritt, Spin Flips and Precession in Black-Hole-Binary Mergers, preprint gr-qc/0612076, available at .
5. Baker J.G., Centrella J., Choi D.-I., Koppitz M., and van Meter J. Gravitational-wave extraction from an inspiraling configuration of merging black holes. Phys. Rev. Lett. 96 (2006) 111102
6. Scheel M.A., Pfeiffer H.P., Lindblom L., Kidder L.E., Rinne O., and Teukolsky S.A. Solving Einstein's equations with dual coordinate frames. Phys. Rev. D 74 (2006) 104006
7. B. Brügmann, J.A. González, M. Hannam, S. Husa, U. Sperhake, Calibration of Moving Puncture Simulations, preprint gr-qc/0610128, available at .
8. Whelan J.T., Krivan W., and Price R.H. Quasi-stationary binary inspiral: II. Radiation-balanced boundary conditions. Class. Quantum Grav. 17 (2000) 4895
9. Whelan J.T., Beetle C., Landry W., and Price R.H. Radiation-balanced simulations for binary inspiral. Class. Quantum Grav. 19 (2002) 1285
10. Price R.H. Binary inspiral: finding the right approximation. Class. Quantum Grav. 21 (2004) S281
11. Andrade Z., Beetle C., Blinov A., Bromley B., Burko L.M., Cranor 1 M., Owen R., and Price R.H. Periodic standing-wave approximation: overview and three-dimensional scalar models. Phys. Rev. D 70 (2004) 064001
12. Bromley B., Owen R., and Price R.H. Periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method. Phys. Rev. D 71 (2005) 104017
13. Torre C.G. The helically-reduced wave equation as a symmetric-positive system. J. Math. Phys. 44 (2003) 6223
14. Torre C.G. Uniqueness of solutions to the helically reduced wave equation with Sommerfeld boundary conditions. J. Math. Phys. 47 (2006) 073501
15. H. Pfeiffer, Initial data for black hole evolutions, Ph.D. Thesis, Cornell University, 2003.
16. Pfeiffer H.P., Kidder L.E., Scheel M.A., and Teukolsky S.A. A multidomain spectral method for solving elliptic equations. Comput. Phys. Commun. 152 (2003) 253
17. M. Boyle, L. Lindblom, H. Pfeiffer, M. Scheel, L.E. Kidder, Testing the Accuracy and Stability of Spectral Methods in Numerical Relativity, preprint gr-qc/0609047, available at http://lanl.arxiv.org/.
18. Beetle C., Bromley B., and Price R.H. Periodic standing-wave approximation: eigenspectral computations for linear gravity and nonlinear toy models. Phys. Rev. D 74 (2006) 024013
19. C. Beetle, B. Bromley, N. Hernández, R.H. Price, The periodic standing-wave approximation: post-Minkowski computations, Phys. Rev. D on 8 August 2007, gr-qc/0708.1141 (submitted for publication).
20. Frankl F.I. On the problems of Chaplygin for mixed subsonic and supersonic flows. Izv. Akad. Nauk SSSR Ser. Mat. 9 (1945) 121
21. {ring operator} ordine di tipo misto. Atti Accad. Naz. Lincei 14 (1923) 133
22. Morawetz C.S. A uniqueness theorem for Frankl's problem. Commun. Pure Appl. Math. 7 (1954) 697
23. Morawetz C.S. Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation. Proc. Roy. Soc. Lond. Ser. A 236 (1956) 141
24. Morawetz C.S. A weak solution for a system of equations of elliptic-hyperbolic type. Commun. Pure Appl. Math. 11 (1958) 315
25. Protter M.H. Uniqueness theorems for the Tricomi problem, I, II. J. Rat. Mech. Anal. 2 (1955) 107
26. Protter M.H. Uniqueness theorems for the Tricomi problem, I, II. J. Rat. Mech. Anal. 4 (1955) 721
27. Murman E.M., and Cole J.D. Calculation of plane steady transonic flows. AIAA J. 9 1 (1971) 114
28. Murman E.M., and Cole J.D. Calculation of plane steady transonic flows. AIAA J. 41 7A (2003) 301 (Reprinted)
29. Caughey D.A., and Jameson A. Development of computational techniques for transonic flows: an historical perspective. Symposium Transsonicum IV vol. 183 (2003), Kluwer Academic, Gottingen
30. Otway T.H. A boundary-value problem for cold plasma dynamics. J. Appl. Math. 3 (2003) 17
31. Bîlǎ N. Application of symmetry analysis to a PDE arising in the car windshield design. SIAM J. Appl. Math. 65 (2004) 113
32. Chaohao G. The mixed PDE for amplifying spiral waves. Lett. Math. Phys. 16 (1988) 69
33. Friedrichs K.O. Symmetric positive linear differential equations. Commun. Pure Appl. Math. 11 (1958) 333
34. M. Jensen, Discontinuous Galerkin methods for Friedrich systems with irregular solutions, PhD Thesis, University of Oxford, 2004.
35. Katsanis T. Numerical solution of symmetric positive differential equations. Math. Comput. 22 (1968) 763
36. Katsanis T. Numerical solution of tricomi equation using theory of symmetric positive differential equations. SIAM J. Numer. Anal. 6 2 (1969) 236
37. Liu J.-L. A finite difference method for symmetric positive differential equations. Math. Comput. 62 (1994) 105
38. Ern A., and Guermond J.-L. Discontinuous Galerkin methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. 44 (2006) 753
39. Ern A., and Guermond J.-L. Discontinuous Galerkin methods for Friedrichs' systems. Part II. Second-order elliptic PDEs. SIAM J. Numer. Anal. 44 (2006) 2363
40. Thompson I.J., and Barnett A.R. Coulomb and Bessel functions of complex arguments and order. J. Comput. Phys. 64 (1986) 490
41. Hagstrom T. Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8 (1999) 47
42. Alpert B., Greengard L., and Hagstrom T. Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal. 37 (2000) 1138
43. Lau S.R. Rapid evaluation of radiation boundary kernels for time-domain wave propagation on blackholes: theory and numerical methods. J. Comput. Phys. 199 (2004) 376
44. Hagstrom T., and Lau S. Radiation boundary conditions for Maxwell's equations: a review of accurate time-domain formulations. J. Comput. Math. 25 (2007) 305-336
45. Coutsias E.A., Hagstrom T., Hesthaven J.S., and Torres D. Integration preconditioners for differential operators in spectral τ-methods. Houston J. Math. 21 (1996) (Special Issue)
46. Coutsias E.A., Hagstrom T., and Torres D. An efficient spectral method for ordinary differential equations with rational function coefficients. Math. Comput. 65 (1996) 611
47. Rivlin T.J. Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory. second ed. (1990), Wiley, New York
48. Beckman P. Orthogonal Polynomials for Engineers and Physicists (1973), Golem Press, Boulder
49. Hesthaven J.S. Integration preconditioning of pseudospectral operators. I. Basic linear operators. SIAM J. Numer. Anal. 35 (1998) 1571
50. von Winckel G., Krishna S., and Coutsias E.A. Spectral element modeling of semiconductor heterostructures. Math. Comput. Model. 43 (2006) 582
51. Kelley C.T. Solving Nonlinear Equations with Newton's Method (2003), SIAM, Philidelphia
52. Adams J.C., and Swarztrauber P.N. SPHEREPACK 3.0: a model development facility. Mon. Weather Rev. 127 (1999) 1872
53. V. Frayssé, L. Giraud, S. Gratton, J. Langou, A set of GMRES routines for real and complex arithmetics on high performance computers, CERFACS Technical Report TR/PA/03/3, public domain software available on , 2003.
54. Demmel J.W. Applied Numerical Linear Algebra (1997), SIAM, Philadelphia
55. Boyd J.P. Chebyshev and Fourier Spectral Methods. second ed. (2001), Dover, New York