Related papers: A Conformal Hyperbolic Formulation of the Einstein…
Motivated by the need to control the exponential growth of constraint violations in numerical solutions of the Einstein evolution equations, two methods are studied here for controlling this growth in general hyperbolic evolution systems.…
The Einstein evolution equations may be written in a variety of equivalent analytical forms, but numerical solutions of these different formulations display a wide range of growth rates for constraint violations. For symmetric hyperbolic…
$[n+1]$-dimensional ($n\geq 3$) smooth Einsteinian spaces of Euclidean and Lorentzian signature are considered. The base manifold $M$ is supposed to be smoothly foliated by a two-parameter family of codimension-two-surfaces which are…
Inspired by a similar analysis for the vacuum conformal Einstein field equations by Paetz [Ann. H. Poincar\'e 16, 2059 (2015)], in this article we show how to construct a system of quasilinear wave equations for the geometric fields…
The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution…
We discuss several explicitly causal hyperbolic formulations of Einstein's dynamical 3+1 equations in a coherent way, emphasizing throughout the fundamental role of the ``slicing function,'' $\alpha$---the quantity that relates the lapse…
We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The…
I present a new, simple method to dynamically control the growth of the discretized constraints during a free evolution of Einstein's equations. During an evolution, any given family of formulations is adjusted off the constraints surface…
Einstein's equations for general relativity, when viewed as a dynamical system for evolving initial data, have a serious flaw: they cannot be proven to be well-posed (except in special coordinates). That is, they do not produce unique…
This paper is devoted to confront two different approaches to the problem of dynam-ical perfect plasticity. Interpreting this model as a constrained boundary value Friedrichs' system enables one to derive admissible hyperbolic boundary…
Motivated by the initial-boundary value problem for the Einstein equations, we propose a definition of symmetric hyperbolicity for systems of evolution equations that are first order in time but second order in space. This can be used to…
We find a choice of variables for the 3+1 formulation of general relativity which casts the evolution equations into (flux-conservative) symmetric-hyperbolic first order form for arbitrary lapse and shift, for the first time. We redefine…
Shibata, Ury\=u and Friedman recently suggested a new decomposition of Einstein's equations that is useful for constructing initial data. In contrast to previous decompositions, the conformal metric is no longer treated as a…
We address nonautonomous initial boundary value problems for decoupled linear first-order one-dimensional hyperbolic systems, investigating the phenomenon of finite time stabilization. We establish sufficient and necessary conditions…
A formulation of Einstein equations is presented that could yield advantages in the study of collisions of binary compact objects during regimes between linear-nonlinear transitions. The key idea behind this formulation is a separation of…
We use partial differential equations (PDEs) to describe physical systems. In general, these equations include evolution and constraint equations. One method used to find solutions to these equations is the Free-evolution approach, which…
This paper is concerned exclusively with axisymmetric spacetimes. We want to develop reductions of Einstein's equations which are suitable for numerical evolutions. We first make a Kaluza-Klein type dimensional reduction followed by an ADM…
A new technique is presented for modifying the Einstein evolution equations off the constraint hypersurface. With this approach the evolution equations for the constraints can be specified freely. The equations of motion for the…
One method of studying the asymptotic structure of spacetime is to apply Penrose's conformal rescaling technique. In this setting, the Einstein equations for the metric and the conformal factor in the unphysical spacetime degenerate where…
Einstein's system of equations in the ADM decomposition involves two subsystems of equations: evolution equations and constraint equations. For numerical relativity, one typically solves the constraint equations only on the initial time…