## Abstract and Applied Analysis

### Constraint-Preserving Boundary Conditions for the Linearized Baumgarte-Shapiro-Shibata-Nakamura Formulation

Alexander M. Alekseenko

#### Abstract

We derive two sets of explicit homogeneous algebraic constraint-preserving boundary conditions for the second-order in time reduction of the linearized Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system. Our second-order reduction involves components of the linearized extrinsic curvature only. An initial-boundary value problem for the original linearized BSSN system is formulated and the existence of the solution is proved using the properties of the reduced system. A treatment is proposed for the full nonlinear BSSN system to construct constraint-preserving boundary conditions without invoking the second order in time reduction. Energy estimates on the principal part of the BSSN system (which is first order in temporal and second order in spatial derivatives) are obtained. Generalizations to the case of nonhomogeneous boundary data are proposed.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 742040, 21 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969171

Digital Object Identifier
doi:10.1155/2008/742040

Mathematical Reviews number (MathSciNet)
MR2407284

Zentralblatt MATH identifier
1148.83305

#### Citation

Alekseenko, Alexander M. Constraint-Preserving Boundary Conditions for the Linearized Baumgarte-Shapiro-Shibata-Nakamura Formulation. Abstr. Appl. Anal. 2008 (2008), Article ID 742040, 21 pages. doi:10.1155/2008/742040. https://projecteuclid.org/euclid.aaa/1220969171

#### References

• A. Anderson and J. W. York Jr., Fixing Einstein's equations,'' Physical Review Letters, vol. 82, no. 22, pp. 4384--4387, 1999.
• T. W. Baumgarte and S. L. Shapiro, Numerical integration of Einstein's field equations,'' Physical Review D, vol. 59, no. 2, Article ID 024007, p. 7, 1999.
• G. Calabrese, L. Lehner, and M. Tiglio, Constraint-preserving boundary conditions in numerical relativity,'' Physical Review D, vol. 65, no. 10, Article ID 104031, p. 13, 2002.
• G. Calabrese, J. Pullin, O. A. Reula, O. Sarbach, and M. Tiglio, Well posed constraint-preserving boundary conditions for the linearized Einstein equations,'' Communications in Mathematical Physics, vol. 240, no. 1-2, pp. 377--395, 2003.
• H. Friedrich and G. Nagy, The initial boundary value problem for Einstein's vacuum field equation,'' Communications in Mathematical Physics, vol. 201, no. 3, pp. 619--655, 1999.
• S. Frittelli and O. A. Reula, First-order symmetric hyperbolic Einstein equations with arbitrary fixed gauge,'' Physical Review Letters, vol. 76, no. 25, pp. 4667--4670, 1996.
• L. E. Kidder, M. A. Scheel, and S. A. Teukolsky, Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations,'' Physical Review D, vol. 64, no. 6, Article ID 064017, p. 13, 2001.
• M. Shibata and T. Nakamura, Evolution of three-dimensional gravitational waves: harmonic slicing case,'' Physical Review D, vol. 52, no. 10, pp. 5428--5444, 1995.
• B. Szilágyi and J. Winicour, Well-posed initial-boundary evolution in general relativity,'' Physical Review D, vol. 68, no. 4, Article ID 041501, p. 5, 2003.
• O. Brodbeck, S. Frittelli, P. Hübner, and O. A. Reula, Einstein's equations with asymptotically stable constraint propagation,'' Journal of Mathematical Physics, vol. 40, no. 2, pp. 909--923, 1999. \setlengthemsep1.7pt
• M. A. Scheel, L. E. Kidder, L. Lindblom, H. P. Pfeiffer, and S. A. Teukolsky, Toward stable 3D numerical evolutions of black-hole spacetimes,'' Physical Review D, vol. 66, no. 12, Article ID 124005, p. 4, 2002.
• M. Holst, L. Lindblom, R. Owen, H. P. Pfeiffer, M. A. Scheel, and L. E. Kidder, Optimal constraint projection for hyperbolic evolution systems,'' Physical Review D, vol. 70, no. 8, Article ID 084017, p. 17, 2004.
• L. Lindblom, M. A. Scheel, L. E. Kidder, H. P. Pfeiffer, D. Shoemaker, and S. A. Teukolsky, Controlling the growth of constraints in hyperbolic evolution systems,'' Physical Review D, vol. 69, no. 12, Article ID 124025, p. 14, 2004.
• M. Alcubierre, G. Allen, B. Brügmann, E. Seidel, and W.-M. Suen, Towards an understanding of the stability properties of the $3+1$ evolution equations in general relativity,'' Physical Review D, vol. 62, no. 12, Article ID 124011, 15 pages, 2000.
• B. Kelly, P. Laguna, K. Lockitch, et al., Cure for unstable numerical evolutions of single black holes: adjusting the standard ADM equations in the spherically symmetric case,'' Physical Review D, vol. 64, no. 8, Article ID 084013, 14 pages, 2001.
• L. E. Kidder, L. Lindblom, M. A. Scheel, L. T. Buchman, and H. P. Pfeiffer, Boundary conditions for the Einstein evolution system,'' Physical Review D, vol. 71, Article ID 064020, 22 pages, 2005.
• D. N. Arnold and N. Tarfulea, Boundary conditions for the Einstein-Christoffel formulation of Einstein's equations,'' in Proceedings of the 6th Mississippi State---UBA Conference on Differential Equations and Computational Simulations, vol. 15 of Electronic Journal of Differential Equations Conference, pp. 11--27, Southwest Texas State University, San Marcos, Tex, USA, 2007.
• C. Gundlach and J. M. Martín-García, Symmetric hyperbolicity and consistent boundary conditions for second-order Einstein equations,'' Physical Review D, vol. 70, no. 4, Article ID 044032, p. 16, 2004.
• S. Frittelli and R. Gómez, Einstein boundary conditions for the $3+1$ Einstein equations,'' Physical Review D, vol. 68, no. 4, Article ID 044014, p. 6, 2003.
• O. Sarbach and M. Tiglio, Boundary conditions for Einstein's field equations: mathematical and numerical analysis,'' Journal of Hyperbolic Differential Equations, vol. 2, no. 4, pp. 839--883, 2005.
• N. Tarfulea, Constraint-preserving boundary conditions for Einstein's equations, Ph.D. dissertation, University of Minnesota, Minneapolis, Minn, USA, 2004.
• C. Gundlach and J. M. Martín-García, Symmetric hyperbolic form of systems of second-order evolution equations subject to constraints,'' Physical Review D, vol. 70, no. 4, Article ID 044031, p. 14, 2004.
• O. Rinne, Stable radiation-controlling boundary conditions for the generalized harmonic Einstein equations,'' Classical and Quantum Gravity, vol. 23, no. 22, pp. 6275--6300, 2006.
• O. A. Reula and O. Sarbach, A model problem for the initial-boundary value formulation of Einstein's field equations,'' Journal of Hyperbolic Differential Equations, vol. 2, no. 2, pp. 397--435, 2005.
• O. Sarbach, Absorbing boundary conditions for Einstein's field equations,'' Journal of Physics, vol. 91, no. 1, Article ID 012005, 15 pages, 2007.
• H.-O. Kreiss and J. Winicour, Problems which are well-posed in a generalized sense with applications to the Einstein equations,'' Classical and Quantum Gravity, vol. 23, no. 16, pp. S405--S420, 2006.
• H.-O. Kreiss, O. A. Reula, O. Sarbach, and J. Winicour, Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates,'' Classical and Quantum Gravity, vol. 24, no. 23, pp. 5973--5984, 2007.
• M. Alcubierre, B. Brügmann, P. Diener, et al., Gauge conditions for long-term numerical black hole evolutions without excision,'' Physical Review D, vol. 67, no. 8, Article ID 084023, p. 18, 2003.
• R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity,'' in Gravitation: An Introduction to Current Research [John Wiley & Sons], pp. 227--265, New York, NY, USA, 1962.
• J. W. York Jr., Kinematics and dynamics of general relativity,'' in Sources of Gravitational Radiation [Cambridge University Press], L. L. Smarr, Ed., pp. 83--126, Cambridge, UK, 1979.
• M. Alcubierre and B. Brügmann, Simple excision of a black hole in $3+1$ numerical relativity,'' Physical Review D, vol. 63, no. 10, Article ID 104006, 6 pages, 2001.
• C. Bona, J. Massó, E. Seidel, and J. Stela, New formalism for numerical relativity,'' Physical Review Letters, vol. 75, no. 4, pp. 600--603, 1995.
• O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations [Academic Press], New York, NY, USA, 1968.
• H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations [Academic Press], vol. 136 of Pure and Applied Mathematics, Boston, Mass, USA, 1989.
• B. Gustafsson, H.-O. Kreiss, and J. Oliger, Time Dependent Problems and Difference Methods [John Wiley & Sons], Pure and Applied MathematicsPure and Applied Mathematics, New York, NY, USA, 1995.
• P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators,'' Communications on Pure and Applied Mathematics, vol. 13, pp. 427--455, 1960.
• P. D. Lax and R. S. Phillips, Scattering theory for dissipative hyperbolic systems,'' Journal of Functional Analysis, vol. 14, pp. 172--235, 1973.
• A. J. Majda, Coercive inequalities for nonelliptic symmetric systems,'' Communications on Pure and Applied Mathematics, vol. 28, pp. 49--89, 1975.
• A. J. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary,'' Communications on Pure and Applied Mathematics, vol. 28, no. 5, pp. 607--675, 1975.
• J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity,'' Transactions of the American Mathematical Society, vol. 291, no. 1, pp. 167--187, 1985.
• B. Bruegmann, W. Tichy, and N. Jansen, Numerical simulation of orbiting black holes,'' Physical Review Letters, vol. 92, no. 21, Article ID 211101, 4 pages, 2004.
• F. John, Partial Differential Equations [Springer], vol. 1 of Applied Mathematical Sciences, New York, NY, USA, 4th edition, 1991.