Related papers: Hyperbolicity of High Order Systems of Evolution E…
A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a…
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…
The causal structure of Einstein's evolution equations is considered. We show that in general they can be written as a first order system of balance laws for any choice of slicing or shift. We also show how certain terms in the evolution…
We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We…
We study hyperbolic systems of one-dimensional partial differential equations under general, possibly non-local boundary conditions. A large class of evolution equations, either on individual 1-dimensional intervals or on general networks,…
After a short introduction to the characteristic geometry underlying weakly hyperbolic systems of partial differential equations we review the notion of symmetric hyperbolicity of first-order systems and that of regular hyperbolicity of…
The constraint equations for smooth $[n+1]$-dimensional (with $n\geq 3$) Riemannian or Lorentzian spaces satisfying the Einstein field equations are considered. It is shown, regardless of the signature of the primary space, that the…
First-order hyperbolic systems are promising as a basis for numerical integration of Einstein's equations. In previous work, the lapse and shift have typically not been considered part of the hyperbolic system and have been prescribed…
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…
We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an…
Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order…
In this paper we study a class of physical systems that combine a finite number of mechanical and thermodynamic observables. We call them finite dimensional thermo-mechanical systems. We introduce these systems by means of simple examples.…
We establish a variant, which has the advantage of introducing only physical characteristics, of the symmetric quasi linear first order system given by H.\ Friedrich for the evolution equations of gravitating fluid bodies in General…
We study strong hyperbolicity of first order partial differential equations for systems with differential constraints. In these cases, the number of equations is larger than the unknown fields, therefore, the standard Kreiss necessary and…
There is a tendency to write the equations of general relativity as a first order symmetric system of time dependent partial differential equations. However, for numerical reasons, it might be advantageous to use a second order formulation…
We review some well posed formulations of the evolution part of the Cauchy problem of General Relativity that we have recently obtained. We include also a new first order symmetric hyperbolic system based directly on the Riemann tensor and…
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…
We investigate the well-posedness of the characteristic initial-boundary value problem for the Einstein equations in Bondi-like coordinates (including Bondi, double-null and affine). We propose a definition of strong hyperbolicity of a…
We introduce a class of singular partial differential equations, the second-order hyperbolic Fuchsian systems, and we investigate the associated initial value problem when data are imposed on the singularity. First of all, we analyze a…
It was shown recently that the constraints on the initial data for Einstein's equations may be posed as an evolutionary problem [9]. In one of the proposed two methods the constraints can be replaced by a first order symmetrizable…