Numerical stability for finite difference approximations of Einstein’s equations. (English) Zbl 1106.65080
Summary: We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in numerical relativity and, more generally, in Hamiltonian formulations of field theories. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We prove stability for certain toy models and the linearized Nagy-Ortiz-Reula formulation of Einstein’s equations. We also find that, unlike in the fully first order case, standard discretizations of some well-posed problems lead to unstable schemes and that the Courant limits are not always simply related to the characteristic speeds of the continuum problem. Finally, we propose methods for testing stability for second order in space hyperbolic systems.
MSC:
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35Q75 | PDEs in connection with 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) |
Keywords:
numerical relativity; finite difference methods; second derivatives; discrete norms; numerical stability; numerical examples; hyperbolic systems; Nagy-Ortiz-Reula formulationReferences:
| [1] | Shibata, M.; Nakamura, T., Phys. Rev. D, 52, 5428 (1995) · Zbl 1250.83027 |
| [2] | Baumgarte, T.; Shapiro, S., Phys. Rev. D, 59, 024007 (1999) · Zbl 1250.83004 |
| [3] | Frittelli, S.; Reula, O., Phys. Rev. Lett., 76, 4667 (1996) |
| [4] | S.D. Hern, Ph.D. Thesis, University of Cambridge, 1999, gr-qc/0004036.; S.D. Hern, Ph.D. Thesis, University of Cambridge, 1999, gr-qc/0004036. |
| [5] | Anderson, A.; York, J. W., Phys. Rev. Lett., 82, 4384 (1999) · Zbl 0949.83012 |
| [6] | Kidder, L. E.; Scheel, M. A.; Teukolsky, S. A., Phys. Rev. D, 64, 064017 (2001) |
| [7] | Sarbach, O.; Calabrese, G.; Pullin, J.; Tiglio, M., Phys. Rev. D, 66, 064002 (2002) |
| [8] | Nagy, G.; Ortiz, O.; Reula, O., Phys. Rev. D, 70, 044012 (2004) |
| [9] | Gundlach, C.; Martin-Garcia, J. M., Phys. Rev. D, 70, 044031 (2004) |
| [10] | Gundlach, C.; Martin-Garcia, J. M., Phys. Rev. D, 70, 044032 (2004) |
| [11] | Beyer, H.; Sarbach, O., Phys. Rev. D, 70, 104004 (2004) |
| [12] | Kreiss, H.; Petersson, N.; Yström, J., SIAM J. Numer. Anal., 40, 1940-1967 (2002) · Zbl 1033.65072 |
| [13] | Szilágyi, B.; Kreiss, H.-O.; Winicour, J., Phys. Rev. D, 71, 104035 (2005) |
| [14] | Richtmyer, R. D.; Morton, K., Difference Methods for Initial Value Problems (1967), Interscience Publisher: Interscience Publisher New York · Zbl 0155.47502 |
| [15] | Frittelli, S.; Gómez, R., J. Math. Phys., 41, 5535 (2000) · Zbl 0977.83006 |
| [16] | Knapp, A.; Walker, E.; Baumgarte, T., Phys. Rev. D, 65, 064031 (2002) |
| [17] | C. Nunn, M.phil. Thesis, University of Southampton, 2005.; C. Nunn, M.phil. Thesis, University of Southampton, 2005. |
| [18] | Arnowitt, R.; Deser, S.; Misner, C., (Witten, L., Gravitation: An Introduction to Current Research (1962), Wiley: Wiley New York) · Zbl 0115.43103 |
| [19] | Bona, C.; Ledvinka, T.; Palenzuela, C.; Žáček, M., Phys. Rev. D, 67, 104005 (2003) |
| [20] | Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time Dependent Problems and Difference Methods (1995), Wiley: Wiley New York · Zbl 0843.65061 |
| [21] | Kreiss, H.-O.; Lorenz, J., Initial-Boundary Value Problems and the Navier-Stokes Equations (1989), Academic Press: Academic Press Boston · Zbl 0689.35001 |
| [22] | Kreiss, H.-O.; Scherer, G., SIAM J. Numer. Anal., 29, 3, 640-646 (1992) · Zbl 0754.65078 |
| [23] | Kreiss, H.-O.; Ortiz, O. E., Lecture Notes in Physics, vol. 604 (2002), Springer: Springer New York |
| [24] | Bona, C.; Ledvinka, T.; Palenzuela, C.; Žáček, M., Phys. Rev. D, 69, 064036 (2004) |
| [25] | M. Babiuc, B. Szilágyi, J. Winicour, gr-qc/0404092.; M. Babiuc, B. Szilágyi, J. Winicour, gr-qc/0404092. |
| [26] | Alcubierre, M., Class. Quantum Grav., 21, 589 (2004) |
| [27] | Szilágyi, B.; Gómez, R.; Bishop, N. T.; Winicour, J., Phys. Rev. D, 62, 104006 (2000) |
| [28] | Calabrese, G.; Pullin, J.; Sarbach, O.; Tiglio, M., Phys. Rev. D, 66, 064011 (2002) |
| [29] | Calabrese, G., Phys. Rev. D, 71, 027501 (2005) |
| [30] | Teukolsky, S. A., Phys. Rev. D, 61, 087501 (2000) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
