161,813 research outputs found
Numerical experiments of adjusted BSSN systems for controlling constraint violations
We present our numerical comparisons between the BSSN formulation widely used
in numerical relativity today and its adjusted versions using constraints. We
performed three testbeds: gauge-wave, linear wave, and Gowdy-wave tests,
proposed by the Mexico workshop on the formulation problem of the Einstein
equations. We tried three kinds of adjustments, which were previously proposed
from the analysis of the constraint propagation equations, and investigated how
they improve the accuracy and stability of evolutions. We observed that the
signature of the proposed Lagrange multipliers are always right and the
adjustments improve the convergence and stability of the simulations. When the
original BSSN system already shows satisfactory good evolutions (e.g., linear
wave test), the adjusted versions also coincide with those evolutions; while in
some cases (e.g., gauge-wave or Gowdy-wave tests) the simulations using the
adjusted systems last 10 times as long as those using the original BSSN
equations. Our demonstrations imply a potential to construct a robust evolution
system against constraint violations even in highly dynamical situations.Comment: to be published in PR
Mixed Hyperbolic - Second-Order Parabolic Formulations of General Relativity
Two new formulations of general relativity are introduced. The first one is a
parabolization of the Arnowitt, Deser, Misner (ADM) formulation and is derived
by addition of combinations of the constraints and their derivatives to the
right-hand-side of the ADM evolution equations. The desirable property of this
modification is that it turns the surface of constraints into a local attractor
because the constraint propagation equations become second-order parabolic
independently of the gauge conditions employed. This system may be classified
as mixed hyperbolic - second-order parabolic. The second formulation is a
parabolization of the Kidder, Scheel, Teukolsky formulation and is a manifestly
mixed strongly hyperbolic - second-order parabolic set of equations, bearing
thus resemblance to the compressible Navier-Stokes equations. As a first test,
a stability analysis of flat space is carried out and it is shown that the
first modification exponentially damps and smoothes all constraint violating
modes. These systems provide a new basis for constructing schemes for long-term
and stable numerical integration of the Einstein field equations.Comment: 19 pages, two column, references added, two proofs of well-posedness
added, content changed to agree with submitted version to PR
Towards an understanding of the stability properties of the 3+1 evolution equations in general relativity
We study the stability properties of the standard ADM formulation of the 3+1
evolution equations of general relativity through linear perturbations of flat
spacetime. We focus attention on modes with zero speed of propagation and
conjecture that they are responsible for instabilities encountered in numerical
evolutions of the ADM formulation. These zero speed modes are of two kinds:
pure gauge modes and constraint violating modes. We show how the decoupling of
the gauge by a conformal rescaling can eliminate the problem with the gauge
modes. The zero speed constraint violating modes can be dealt with by using the
momentum constraints to give them a finite speed of propagation. This analysis
sheds some light on the question of why some recent reformulations of the 3+1
evolution equations have better stability properties than the standard ADM
formulation.Comment: 15 pages, 9 figures. Added a new section, plus incorporated many
comments made by refere
Stochastic model predictive control for constrained networked control systems with random time delay
In this paper the continuous time stochastic constrained optimal control problem is formulated for the class of networked control systems assuming that time delays follow a discrete-time, finite Markov chain . Polytopic overapproximations of the system's trajectories are employed to produce a polyhedral inner approximation of the non-convex constraint set resulting from imposing the constraints in continuous time. The problem is cast in a Markov jump linear systems (MJLS) framework and a stochastic MPC controller is calculated explicitly, oine, coupling dynamic programming with parametric piecewise quadratic (PWQ) optimization. The calculated control law leads to stochastic stability of the closed loop system, in the mean square sense and respects the state and input constraints in continuous time
Stability Analysis of a Rigid Body with Attached Geometrically Nonlinear Rod by the Energy-Momentum Method
This paper applies the energy-momentum method to the problem of nonlinear stability of relative equilibria of a rigid body with attached flexible appendage in a uniformly rotating state. The appendage is modeled as a geometrically exact rod which allows for finite bending, shearing and twist in three dimensions. Application of the energy-momentum method to this example depends crucially on a
special choice of variables in terms of which the second variation block diagonalizes into blocks associated with rigid body modes and internal vibration modes respectively. The analysis yields a nonlinear stability result which states that relative equilibria are nonlinearly stable provided that; (i) the angular velocity is bounded above by the square root of the minimum eigenvalue of an associated
linear operator and, (ii) the whole assemblage is rotating about the minimum axis of inertia
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