Related papers: Cauchy-characteristic extraction in numerical rela…
The goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size $\epsilon$, is almost…
Although there are many results on the global solvability and the precise description of the large time behaviors of solutions to the initial-boundary value problems of the one-dimensional viscous radiative and reactive gas in bounded…
This paper studies the Cauchy problem for systems of semi-linear wave equations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial…
In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. We prove that given initial data on a maximal compact spacelike hypersurface $\Sigma \simeq \overline{B(0,1)} \subset \mathbb{R}^3$ and…
We analyze existence and properties of solutions of two-dimensional general relativistic initial data sets with a negative cosmological constant, both on spacelike and characteristic surfaces. A new family of such vacuum, spacelike data…
We establish several results on gluing/embedding/extending geometric structures in vacuum spacetimes with a cosmological constant in any spacetime dimensions $d\ge 4$, with emphasis on characteristic data. A useful tool is provided by the…
We consider the Cauchy problem for the wave equation in $\Omega\times{\mathbb R}$ with data given on some part of the boundary $\partial\Omega\times{\mathbb R}$. We provide a reconstruction algorithm for this problem based on analytic…
We prove identification of coefficients up to gauge by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of $\mathbb{C}$. In the geometric setting, we fix a Riemann surface with boundary,…
The Cauchy+characteristic matching (CCM) problem for the scalar wave equation is investigated in the background geometry of a Schwarzschild black hole. Previously reported work developed the CCM framework for the coupled…
We consider the strategy of realizing the solution of a Cauchy problem with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge…
This thesis is concerned with the development and application of conformal techniques to numerical calculations of asymptotically flat spacetimes. The conformal compactification technique enables us to calculate spatially unbounded domains,…
We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and…
We develop a numerical method suitable for gravitational collapse based on Cauchy evolution with an ingoing characteristic boundary. Unlike similar methods proposed recently (Ripley; Bieri, Garfinkle & Yau 2019/20), the numerical grid…
The asymptotic behavior of geometry near the boundary of maximal Cauchy development is studied using a perturbative method, which at the zeroth order reduces Einstein's equations to an exactly solvable set of equations---Einstein's…
We present a spectral algorithm for solving the full nonlinear vacuum Einstein field equations in the Bondi framework. Developed within the Spectral Einstein Code (SpEC), we demonstrate spectral characteristic evolution as a technical…
Although the traditional form of the Einstein field equations is intrinsically four-dimensional, the field of numerical general relativity focuses on the reformulation of these equations as a 3 + 1-dimensional Cauchy problem, in which…
In recent years, the Characteristic formulation of numerical relativity has found increasing use in the extraction of gravitational radiation from numerically generated spacetimes. In this paper, we formulate the Characteristic initial…
Consider a real-analytic orientable connected complete Riemannian manifold $M$ with boundary of dimension $n\ge 2$ and let $k$ be an integer $1\le k\le n$. In the case when $M$ is compact of dimension $n\ge 3$, we show that the manifold and…
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional…
The general class of Robinson-Trautman metrics that describe gravitational radiation in the exterior of bounded sources in four space-time dimensions is shown to admit zero curvature formulation in terms of appropriately chosen…