1,079 research outputs found
The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves
The null-timelike initial-boundary value problem for a hyperbolic system of
equations consists of the evolution of data given on an initial characteristic
surface and on a timelike worldtube to produce a solution in the exterior of
the worldtube. We establish the well-posedness of this problem for the
evolution of a quasilinear scalar wave by means of energy estimates. The
treatment is given in characteristic coordinates and thus provides a guide for
developing stable finite difference algorithms. A new technique underlying the
approach has potential application to other characteristic initial-boundary
value problems.Comment: Version to appear in Class. Quantum Gra
Boundary conditions for coupled quasilinear wave equations with application to isolated systems
We consider the initial-boundary value problem for systems of quasilinear
wave equations on domains of the form , where is
a compact manifold with smooth boundaries . By using an
appropriate reduction to a first order symmetric hyperbolic system with maximal
dissipative boundary conditions, well posedness of such problems is established
for a large class of boundary conditions on . We show that our
class of boundary conditions is sufficiently general to allow for a well posed
formulation for different wave problems in the presence of constraints and
artificial, nonreflecting boundaries, including Maxwell's equations in the
Lorentz gauge and Einstein's gravitational equations in harmonic coordinates.
Our results should also be useful for obtaining stable finite-difference
discretizations for such problems.Comment: 22 pages, no figure
The Initial-Boundary Value Problem in General Relativity
In this article we summarize what is known about the initial-boundary value
problem for general relativity and discuss present problems related to it.Comment: 11 pages, 2 figures. Contribution to a special volume for Mario
Castagnino's seventy fifth birthda
Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates
In recent work, we used pseudo-differential theory to establish conditions
that the initial-boundary value problem for second order systems of wave
equations be strongly well-posed in a generalized sense. The applications
included the harmonic version of the Einstein equations. Here we show that
these results can also be obtained via standard energy estimates, thus
establishing strong well-posedness of the harmonic Einstein problem in the
classical sense.Comment: More explanatory material and title, as will appear in the published
article in Classical and Quantum Gravit
Strongly hyperbolic second order Einstein's evolution equations
BSSN-type evolution equations are discussed. The name refers to the
Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution
equations, without introducing the conformal-traceless decomposition but
keeping the three connection functions and including a densitized lapse. It is
proved that a pseudo-differential first order reduction of these equations is
strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner
evolution equations are found to be weakly hyperbolic. In both cases, the
positive densitized lapse function and the spacelike shift vector are arbitrary
given fields. This first order pseudodifferential reduction adds no extra
equations to the system and so no extra constraints.Comment: LaTeX, 16 pages, uses revtex4. Referee corections and new appendix
added. English grammar improved; typos correcte
Blowup of small data solutions for a quasilinear wave equation in two space dimensions
For the quasilinear wave equation
\partial_t^2u - \Delta u = u_t u_{tt},
we analyze the long-time behavior of classical solutions with small (not
rotationally invariant) data. We give a complete asymptotic expansion of the
lifespan and describe the solution close to the blowup point. It turns out that
this solution is a ``blowup solution of cusp type,'' according to the
terminology of the author.Comment: 31 pages, published versio
Some mathematical problems in numerical relativity
The main goal of numerical relativity is the long time simulation of highly
nonlinear spacetimes that cannot be treated by perturbation theory. This
involves analytic, computational and physical issues. At present, the major
impasses to achieving global simulations of physical usefulness are of an
analytic/computational nature. We present here some examples of how analytic
insight can lend useful guidance for the improvement of numerical approaches.Comment: 17 pages, 12 graphs (eps format
Harmonic Initial-Boundary Evolution in General Relativity
Computational techniques which establish the stability of an
evolution-boundary algorithm for a model wave equation with shift are
incorporated into a well-posed version of the initial-boundary value problem
for gravitational theory in harmonic coordinates. The resulting algorithm is
implemented as a 3-dimensional numerical code which we demonstrate to provide
stable, convergent Cauchy evolution in gauge wave and shifted gauge wave
testbeds. Code performance is compared for Dirichlet, Neumann and Sommerfeld
boundary conditions and for boundary conditions which explicitly incorporate
constraint preservation. The results are used to assess strategies for
obtaining physically realistic boundary data by means of Cauchy-characteristic
matching.Comment: 31 pages, 14 figures, submitted to Physical Review
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