Related papers: Will hyperbolic formulations help numerical relati…
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…
In this paper, we generalize the well-known hyperbolic numbers to certain numeric structures scaled by the real numbers. Under our scaling of $\mathbb{R}$, the usual hyperbolic numbers are understood to be our 1-scaled hyperbolic numbers.…
Systems of PDEs comprised of a combination of constraints and evolution equations are ubiquitous in physics. For both theoretical and practical reasons, such as numerical integration, it is desirable to have a systematic understanding of…
Building upon previous works by Young, Chernov-Zhang and Bruin-Melbourne-Terhesiu, we present a general scheme to improve bounds on the statistical properties (in particular, decay of correlations, and rates in the almost sure invariant…
An averaging method for getting uniformly valid asymptotic approximations of the solution of hyperbolic systems of equations is presented. The averaged system of equations disintegrates into independent equations for non-resonance systems.…
Dimensional reductions of various higher dimensional (super)gravity theories lead to effectively two-dimensional field theories described by gravity coupled G/H nonlinear sigma-models. We show that a new set of complexified variables can be…
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…
This article begins with a brief introduction to numerical relativity aimed at readers who have a background in applied mathematics but not necessarily in general relativity. I then introduce and summarise my work on the problem of treating…
We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative…
The canonical approach to Riemannian quantum gravity is reviewed with reference to local supersymmetry, to the classical boundary-value problem arising from the Hartle-Hawking quantum state, and particularly for (anti-)self-dual geometries.…
Outer boundary conditions for strongly and symmetric hyperbolic formulations of 3D Einstein's field equations with a live gauge condition are discussed. The boundary conditions have the property that they ensure constraint propagation and…
We study the properties of the principal symbol of the 3+1 equations of motion in Teleparallel Equivalent of General Relativity (TEGR) and assess the conditions for hyperbolicity. We use the Hamiltonian formulation based on the vectorial,…
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…
Systematic numerical investigations of the asymptotics of near Schwarzschild vacuum initial data sets is carried out by inspecting solutions to the parabolic-hyperbolic and to the algebraic-hyperbolic forms of the constraints, respectively.…
Bondi-like (single-null) characteristic formulations of general relativity are used for numerical work in both asymptotically flat and anti-de Sitter spacetimes. Well-posedness of the resulting systems of partial differential equations,…
In this paper, we deal with generalizations of real Einstein numbers to various spaces and dimensions. We search operations and their properties in generalized settings. Especially, we are interested in the generalized operation of…
We show that, even for extremely stiff systems, explicit integration may compete in both accuracy and speed with implicit methods if algebraic methods are used to stabilize the numerical integration. The required stabilizing algebra depends…
The relation between the Husain-Kucha\v{r} model and some extensions thereof that incorporate a dynamical time variable is explored. To this end, we rely on the geometric approach to the Hamiltonian dynamics of singular (constrained)…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
The theory of symmetric-hyperbolic systems is useful for constructing smooth solutions of nonlinear wave equations, and for studying their singularities, including shock waves. We present the main techniques which are required to apply the…