60 research outputs found
Diagonalizability of Constraint Propagation Matrices
In order to obtain stable and accurate general relativistic simulations,
re-formulations of the Einstein equations are necessary. In a series of our
works, we have proposed using eigenvalue analysis of constraint propagation
equations for evaluating violation behavior of constraints. In this article, we
classify asymptotical behaviors of constraint-violation into three types
(asymptotically constrained, asymptotically bounded, and diverge), and give
their necessary and sufficient conditions. We find that degeneracy of
eigenvalues sometimes leads constraint evolution to diverge (even if its
real-part is not positive), and conclude that it is quite useful to check the
diagonalizability of constraint propagation matrices. The discussion is general
and can be applied to any numerical treatments of constrained dynamics.Comment: 4 pages, RevTeX, one figure, added one paragraph in concluding
remarks. The version to appear in Class. Quant. Grav. (Lett
Constraint propagation in N+1-dimensional space-time
Higher dimensional space-time models provide us an alternative interpretation
of nature, and give us different dynamical aspects than the traditional
four-dimensional space-time models. Motivated by such recent interests,
especially for future numerical research of higher-dimensional space-time, we
study the dimensional dependence of constraint propagation behavior. The
Arnowitt-Deser-Misner evolution equation has matter terms which depend on ,
but the constraints and constraint propagation equations remain the same. This
indicates that there would be problems with accuracy and stability when we
directly apply the ADM formulation to numerical simulations as we have
experienced in four-dimensional cases. However, we also conclude that previous
efforts in re-formulating the Einstein equations can be applied if they are
based on constraint propagation analysis.Comment: 4 pages, to appear in Gen. Rel. Gra
Asymptotically constrained and real-valued system based on Ashtekar's variables
We present a set of dynamical equations based on Ashtekar's extension of the
Einstein equation. The system forces the space-time to evolve to the manifold
that satisfies the constraint equations or the reality conditions or both as
the attractor against perturbative errors. This is an application of the idea
by Brodbeck, Frittelli, Huebner and Reula who constructed an asymptotically
stable (i.e., constrained) system for the Einstein equation, adding dissipative
forces in the extended space. The obtained systems may be useful for future
numerical studies using Ashtekar's variables.Comment: added comments, 6 pages, RevTeX, to appear in PRD Rapid Com
Constraint Propagation of -adjusted Formulation - Another Recipe for Robust ADM Evolution System
With a purpose of constructing a robust evolution system against numerical
instability for integrating the Einstein equations, we propose a new
formulation by adjusting the ADM evolution equations with constraints. We apply
an adjusting method proposed by Fiske (2004) which uses the norm of the
constraints, C2. One of the advantages of this method is that the effective
signature of adjusted terms (Lagrange multipliers) for constraint-damping
evolution is pre-determined. We demonstrate this fact by showing the
eigenvalues of constraint propagation equations. We also perform numerical
tests of this adjusted evolution system using polarized Gowdy-wave propagation,
which show robust evolutions against the violation of the constraints than that
of the standard ADM formulation.Comment: 11 pages, 5 figures. To be published in Phys. Rev.
Constructing hyperbolic systems in the Ashtekar formulation of general relativity
Hyperbolic formulations of the equations of motion are essential technique
for proving the well-posedness of the Cauchy problem of a system, and are also
helpful for implementing stable long time evolution in numerical applications.
We, here, present three kinds of hyperbolic systems in the Ashtekar formulation
of general relativity for Lorentzian vacuum spacetime. We exhibit several (I)
weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric
hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's
original equations form a weakly hyperbolic system. We discuss how gauge
conditions and reality conditions are constrained during each step toward
constructing a symmetric hyperbolic system.Comment: 15 pages, RevTeX, minor changes in Introduction. published as Int. J.
Mod. Phys. D 9 (2000) 1
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