Initial data for black hole-neutron star binaries: a flexible, high-accuracy spectral method

We present a new numerical scheme to solve the initial value problem for black hole-neutron star binaries. This method takes advantage of the flexibility and fast convergence of a multidomain spectral representation of the initial data to construct high-accuracy solutions at a relatively low computational cost. We provide convergence tests of the method for both isolated neutron stars and irrotational binaries. In the second case, we show that we can resolve the small inconsistencies that are part of the quasi-equilibrium formulation, and that these inconsistencies are significantly smaller than observed in previous works. The possibility of generating a wide variety of initial data is also demonstrated through two new configurations inspired by results from binary black holes. First, we show that choosing a modified Kerr-Schild conformal metric instead of a flat conformal metric allows for the construction of quasi-equilibrium binaries with a spinning black hole. Second, we construct binaries in low-eccentricity orbits, which are a better approximation to astrophysical binaries than quasi-equilibrium systems.Comment: 19 pages, 11 figures, Modified to match final PRD versio

Isometric bending requires local constraints on free edges

While the shape equations describing the equilibrium of an unstretchable thin sheet that is free to bend are known, the boundary conditions that supplement these equations on free edges have remained elusive. Intuitively, unstretchability is captured by a constraint on the metric within the bulk. Naively one would then guess that this constraint is enough to ensure that the deformations determining the boundary conditions on these edges respect the isometry constraint. If matters were this simple, unfortunately, it would imply unbalanced torques (as well as forces) along the edge unless manifestly unphysical constraints are met by the boundary geometry. In this paper we identify the source of the problem: not only the local arc-length but also the geodesic curvature need to be constrained explicitly on all free edges. We derive the boundary conditions which follow. Contrary to conventional wisdom, there is no need to introduce boundary layers. This framework is applied to isolated conical defects, both with deficit as well, but more briefly, as surplus angles. Using these boundary conditions, we show that the lateral tension within a circular cone of fixed radius is equal but opposite to the radial compression, and independent of the deficit angle itself. We proceed to examine the effect of an oblique outer edge on this cone perturbatively demonstrating that both the correction to the geometry as well as the stress distribution in the cone kicks in at second order in the eccentricity of the edge.Comment: 25 pages, 3 figure

An Efficient Method For Solving Highly Anisotropic Elliptic Equations

Solving elliptic PDEs in more than one dimension can be a computationally expensive task. For some applications characterised by a high degree of anisotropy in the coefficients of the elliptic operator, such that the term with the highest derivative in one direction is much larger than the terms in the remaining directions, the discretized elliptic operator often has a very large condition number - taking the solution even further out of reach using traditional methods. This paper will demonstrate a solution method for such ill-behaved problems. The high condition number of the D-dimensional discretized elliptic operator will be exploited to split the problem into a series of well-behaved one and (D-1)-dimensional elliptic problems. This solution technique can be used alone on sufficiently coarse grids, or in conjunction with standard iterative methods, such as Conjugate Gradient, to substantially reduce the number of iterations needed to solve the problem to a specified accuracy. The solution is formulated analytically for a generic anisotropic problem using arbitrary coordinates, hopefully bringing this method into the scope of a wide variety of applications.Comment: 37 pages, 11 figure

Plateau angle conditions for the vector-valued Allen-Cahn equation

Under proper hypotheses, we rigorously derive the Plateau angle conditions at triple junctions of diffused interfaces in three dimensions, starting from the vector-valued Allen-Cahn equation with a triple-well potential. Our derivation is based on an application of the divergence theorem using the divergence-free form of the equation via an associated stress tensor.Comment: 15 pages, 1 figure; minor revision, added proof

Isolated and dynamical horizons and their applications

Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss them have varied considerably from one area to another. Over the last five years a new, quasi-local framework was introduced to analyze diverse facets of black holes in a unified manner. In this framework, evolving black holes are modeled by dynamical horizons and black holes in equilibrium by isolated horizons. We review basic properties of these horizons and summarize applications to mathematical physics, numerical relativity and quantum gravity. This paradigm has led to significant generalizations of several results in black hole physics. Specifically, it has introduced a more physical setting for black hole thermodynamics and for black hole entropy calculations in quantum gravity; suggested a phenomenological model for hairy black holes; provided novel techniques to extract physics from numerical simulations; and led to new laws governing the dynamics of black holes in exact general relativity.Comment: 77 pages, 12 figures. Typos and references correcte

Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with appropriate predictor-corrector method to achieve higher resolution. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed, thus unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to Geosciences. Other author's papers can be downloaded at http://www.denys-dutykh.com

Density and flux distributions of neutral gases in the lunar atmosphere

Neon, argon, and helium density and flux distributions in lunar atmospher

GRMHD in axisymmetric dynamical spacetimes: the X-ECHO code

We present a new numerical code, X-ECHO, for general relativistic magnetohydrodynamics (GRMHD) in dynamical spacetimes. This is aimed at studying astrophysical situations where strong gravity and magnetic fields are both supposed to play an important role, such as for the evolution of magnetized neutron stars or for the gravitational collapse of the magnetized rotating cores of massive stars, which is the astrophysical scenario believed to eventually lead to (long) GRB events. The code is based on the extension of the Eulerian conservative high-order (ECHO) scheme [Del Zanna et al., A&A 473, 11 (2007)] for GRMHD, here coupled to a novel solver for the Einstein equations in the extended conformally flat condition (XCFC). We fully exploit the 3+1 Eulerian formalism, so that all the equations are written in terms of familiar 3D vectors and tensors alone, we adopt spherical coordinates for the conformal background metric, and we consider axisymmetric spacetimes and fluid configurations. The GRMHD conservation laws are solved by means of shock-capturing methods within a finite-difference discretization, whereas, on the same numerical grid, the Einstein elliptic equations are treated by resorting to spherical harmonics decomposition and solved, for each harmonic, by inverting band diagonal matrices. As a side product, we build and make available to the community a code to produce GRMHD axisymmetric equilibria for polytropic relativistic stars in the presence of differential rotation and a purely toroidal magnetic field. This uses the same XCFC metric solver of the main code and has been named XNS. Both XNS and the full X-ECHO codes are validated through several tests of astrophysical interest.Comment: 18 pages, 9 figures, accepted for publication in A&