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The main topic of the workshop is code comparison in numerical relativity but we welcome anyone with analytic expertise in any of the underlying issues affecting numerical stability: hyperbolic formulations of Einstein's equations,
boundary conditions, constraint preservation, gauge conditions. Talks of such general interest will be scheduled on the first three or four days of the workshop. Contact Manuel Tiglio (tiglio_at_astro.cornell.edu), Oscar Reula (reula_at_fis.uncor.edu) or Jeff Winicour (jeff_at_einstein.phyast.pitt.edu) to be put on the schedule.
The talks announced so far for this section are the following (in alphabetical order).

  1. Tilman Vogel, Tuesday 11:00A
    Title:   Stability and constraint propagation

    Authors: Tilman Vogel,
              AEI, Golm
         and: Joerg Frauendiener,
              Institut für Astronomie und Astrophysik, Universität Tübingen

    Abstract: A typical problem of numerical relativity is the instable
    behaviour of simulations. Growing constraint quantities destroy the
    physical properties of the solutions on short time-scales.

    Two main sources are connected to the formulation of the IBVP: (i)
    Inconsistent boundary conditions inject unphysical data into the
    numerical domain. (ii) Small but unavoidable constraint violations in
    the initial data trigger instabilities of the evolution equations. The
    latter problem, which is already part of the continuum formulation, will
    be discussed in this talk.

    It will be sketched, how a stability analysis for such perturbations can
    be carried out. The essential influence of the choice of the foliation
    will be demonstrated. As examples, analytical results for the Maxwell
    equations and the Bianchi equation will be presented.

    [1] Joerg Frauendiener and Tilman Vogel: On the stability of constraint
    propagation, gr-qc/0410100


  2. Bernd Reimann. Tuesday 11:00AM

    Instituto de Ciencias Nucleares, Universidad Nacional
    Aut{\'o}noma de M{\'e}xico, A.P. 70-543, M{\'e}xico D.F. 04510, M{\'e}xico

    Max Planck Institut f\"ur Gravitationsphysik,
    Albert Einstein Institut, Am M\"uhlenberg 1, 14476 Golm, Germany

    Title: Avoiding Gauge and Constraint Shocks in 1D Numerical Relativity

    Abstract:
    I will report on recent work (gr-qc/0411094) done together with M. Alcubierre,
    J.A. Gonzalez and D. Nunez:

    For evolution systems of the type found in Numerical Relativity we successfully
    make use of mathematical theory regarding the singularity formation in wavelike
    hyperbolic equations. This theory states that for systems of partial
    differential equations two kinds of shocks can be expected to form, namely
    "gradient catastrophes" and "blow-ups within finite time". For both mechanisms
    we are able to derive shock-avoiding conditions which we refer to as "indirect
    linear degeneracy" and "source criteria". For a toy problem, the wave equation
    with dynamic wave speed and quadratic source terms, we show that the solution
    has wavelike and shock-avoiding character only if both conditions are
    satisfied.

    In our primary example, Einstein's equations in spherical symmetry, the two
    criteria detect "gauge and constraint shocks". Here the gauge shocks are
    associated with the foliation itself, and the shock formation in the evolution
    variables can be avoided by choosing the lapse and the shift function
    appropriately. The constraint shocks arise due to numerically unfavorable terms
    in the evolution equations and cause constraint violations to become infinite
    within finite time. For the ADM system in spherical symmetry, we show that
    these shocks can be avoided by adding constraints to some evolution equations.
    By doing so, we obtain a 1-parameter family of shock-avoiding evolution systems
    which in numerical experiments such as the robust stability test shows excellent
    numerical behavior.

  3. Osvaldo Moreschi, Tuesday 3PM
    FaMAF, Universidad de Cordoba, Argentina.

    Ttitle: An interior-asymptotics approach to the binary black hole problem

    author: Osvaldo M. Moreschi

    abstract:
    We present an approach to the binary black hole problem that makes
    use of two complementary techniques.

    We study the dynamical equations for a binary system of compact
    objects arising from the asymptotic structure of the corresponding
    isolated system. We make use of the center of mass frame at
    future null infinity and keep terms up to first order of
    gravitational radiation.

    We complement this system with the equations of motion of two
    black holes, in the interior of the spacetime.


  4. Marronetti, Pedro, Wednesday 9:30AM

    Florida Atlantic University


    Title
    : "Binary Neutron Star Simulations as Testing Ground
    for Numerical Relativity".

    Abstract
    : New formulations for the evolution of gravitational fields in numerical

    relativity are continuously presented to the scientific community. Their

    main goal is to achieve stable and reliable evolutions of compact-object

    binaries. However, due to the complexity of the required numerical work,

    few of the many formulations found in the literature have been tested on

    binary evolutions. We introduce a new testing ground for numerical

    methods based on the simulation of binary neutron stars (BNS). Our

    objective is to benchmark new formalisms against the currently most

    stable simulations. BNS simulations usually require extensive

    computational resources and the length of the runs could, in principle,

    render these tests impractical. We show how small, low resolution grids

    can be used to gain insight into the stability of different numerical

    schemes, with runs that only take a few hours on single-processor

    workstations.

  5. Carles Bona, Wednesday 11AM

    UIB, Palma-Spain

    Title: "Covariant freezing shift conditions"

    The well known 'gamma-freezing' shift condition is generalized in a
    covariant way. Two lapse conditions are also considered in this
    context. The relationship of these gauge conditions with the existence
    of a time Killing vector is pointed out. Two simple test cases are
    investigated numerically: the 1D Kerr-Schild Black Hole and the
    Minkowsky gauge waves.


  6. Sascha Husa, Wednesday 3PM
    AEI
    TBA


  7. Santiago Gomez, Wednesday 4PM
    FaMAF, Cordoba.

    Title: Dynamics of tetrad fields of Schwarzchild spacetime


     In tetrad formalism, the Einstein equations can be reduced to a simple
    system of ordinary differential equations. The system is made up of 30
    scalar fields, in addition there are 30 constrains to be satisfied by the
    system. The evolution of the constrains have the same stability property
    as the original fields. The gauge data is provided by the temporal
    projection of the four tetrad vectors and six connections vectors. The
    system have a natural unstable behavior unless a trivial and special gauge
    is taken, when the temporal projection of the connection are assumed to be
    null. Only for static frames we found a way of having a system with stable
    behavior without that special and trivial gauge, it consists of changing
    the sign of the positive eigenvalues of the system. Into the evolution
    equation of the fields appears some components of the Weyl tensor with the
    last two indices contracted with the tetrad, they are fixed by calculing
    them for the case of Schwarzschild spacetime. The initial data for all
    fields are obteined in the same way. The angular part of the tetrad will
    be separated into six diferent patches using cubito coordinate. Due the
    spherical symmetry of the system, all patches will have the same
    behavior.
    In a near future we expect to be ready to evolve non-spherical symmetric
    spacetimes into this formalism.





  8. Jeff Winicour, Wednesday 4:30PM
    Pittsburgh University and AEI

    Shifted Gauge Wave Simulations

There will also be presentations by representatives of the groups carrying out code comparison. These talks will explain the formalism underlying the code, its numerical implementation, and its results on the standardized tests. At
present, the following people have committed to presenting such talks:

  1. Carles Bona
  2. Sascha Husa
  3. Carlos Palenzuela
  4. Denis Pollney
  5. Bela Szilagyi
  6. Manuel Tiglio
  7. Jeff Winicour

The second week of the workshop will be devoted to carrying out code tests and comparisons and to the design of further tests. The code comparisons will focus on:

  • The first round of Mexico tests with periodic boundary conditions, with
    the proposed  modifications proposed to run at higher amplitudes and with
  • Runge-Kutta integrators as replacement for ICN. 
  • Mexico tests for a gauge wave with shift.
  • The first round of boundary tests.
  • The design of new tests will focus on black hole spacetimes.



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