We present SphericalNR, a new framework for the publicly available Einstein Toolkit that numerically solves the Einstein field equations coupled to the equations of general relativistic magnetohydrodynamics (GRMHD) in a 3+1 split of spacetime in spherical coordinates without symmetry assumptions. The spacetime evolution is performed using reference-metric versions of either the Baumgarte-Shapiro-Shibata-Nakamura equations or the fully covariant and conformal Z4 system with constraint damping. We have developed a reference-metric version of the Valencia formulation of GRMHD with a vector potential method, guaranteeing the absence of magnetic monopoles during the evolution. In our framework, every dynamical field (both spacetime and matter) is evolved using its components in an orthonormal basis with respect to the spherical reference-metric. Furthermore, all geometric information about the spherical coordinate system is encoded in source terms appearing in the evolution equations. This allows for the straightforward extension of Cartesian high-resolution shock-capturing finite volume codes to use spherical coordinates with our framework. To this end, we have adapted GRHydro, a Cartesian finite volume GRMHD code already available in the Einstein Toolkit, to use spherical coordinates. We present the full evolution equations of the framework, as well as details of its implementation in the Einstein Toolkit. We validate SphericalNR by demonstrating it passes a variety of challenging code tests in static and dynamical spacetimes.
Numerical relativity codes that do not make assumptions on spatial symmetries most commonly adopt Cartesian coordinates. While these coordinates have many attractive features, spherical coordinates are much better suited to take advantage of approximate symmetries in a number of astrophysical objects, including single stars, black holes and accretion disks. While the appearance of coordinate singularities often spoils numerical relativity simulations in spherical coordinates, especially in the absence of any symmetry assumptions, it has recently been demonstrated that these problems can be avo…
The production of numerical relativity waveforms that describe quasicircular binary black hole mergers requires high-quality initial data, and an algorithm to iteratively reduce residual eccentricity. To date, these tools remain closed source, or in commercial software that prevents their use in high performance computing platforms. To address these limitations, and to ensure that the broader numerical relativity community has access to these tools, herein we provide all the required elements to produce high-quality numerical relativity simulations in supercomputer platforms, namely: open sour…
The Einstein Toolkit represents a unique opportunity for students to explore the world of numerical relativity, without the need for high-level computing power or knowledge of the mathematics behind the simulations themselves. This document aims to act as a resource for students aiming to learn about and get involved with Numerical Relativity, focusing specifically on the modelling of Binary Black Hole Mergers in order to produce sample Gravitational Waveforms as would be measured by Gravitational Wave detectors such as LIGO. The feasibility of using the Einstein Toolkit on Amazon's Web Servic…
Sector Models - A Toolkit for Teaching General Relativity. Part 1: Curved Spaces and Spacetimes
向高中生或本科生教授广义相对论必须基于概念而不是数学的方法。在本文中，我们提出了这样一种方法，只需要初等数学。介绍广义相对论的中心思想是使用所谓的扇形模型。扇形模型用Regge演算的方法描述弯曲的空间，用欧几里得几何将其分成块。这个过程类似于用平面三角形逼近曲面。我们概述了高中和本科生的研讨会，介绍了弯曲空间的概念，通过部门模型的黑洞。我们进一步描述了弯曲时空扇形模型的扩展。时空模型适用于具有狭义相对论基础知识的学习者。有关在线教学材料，请参阅http://www.spacetimetravel.org。--我们将在这张桌子上放一张桌子。Das zentrale Werkzeug unserer Einführung sind sogenannte Sektormodelle，die gekrümmte Räbrim Sinne des Regge-Kalküls durch eine Zerlegung in kleine，ungekrümmte Sektoren beschreiben，ähnlich der Triangulierung einer gekrümmten Fläche.我们将为Schüler/innen und Studierende举办一个研讨会，在Sektormodellen Schwarzer Löcher eingeführt werden上进行演讲。…
Sector models are tools that make it possible to teach the basic principles of the general theory of relativity without going beyond elementary mathematics. This contribution shows how sector models can be used to determine geodesics. We outline a workshop for high school and undergraduate students that addresses gravitational light deflection by means of the construction of geodesics on sector models. Geodesics close to a black hole are used by way of example. The contribution also describes a simplified calculation of sector models that students can carry out on their own. The accuracy of th…
Sector models permit a model-based approach to the general theory of relativity. The approach has its focus on the geometric concepts and uses no more than elementary mathematics. This contribution shows how to construct the paths of light and free particles on a spacetime sector model. Radial paths close to a black hole are used by way of example. We outline two workshops on gravitational redshift and on vertical free fall, respectively, that we teach for undergraduate students. The contribution also describes a simplified calculation of the spacetime sector model that students can carry out …
Numerical simulations of Einstein's field equations provide unique insights into the physics of compact objects moving at relativistic speeds, and which are driven by strong gravitational interactions. Numerical relativity has played a key role to firmly establish gravitational wave astrophysics as a new field of research, and it is now paving the way to establish whether gravitational wave radiation emitted from compact binary mergers is accompanied by electromagnetic and astro-particle counterparts. As numerical relativity continues to blend in with routine gravitational wave data analyses t…
Ameliorating the Courant-Friedrichs-Lewy condition in spherical coordinates: A double FFT filter method for general relativistic MHD in dynamical spacetimes
A Reproducible Black Hole-Neutron Star Merger Gallery Example for the Einstein Toolkit
University of Southampton（南安普顿大学）；University of British Columbia（不列颠哥伦比亚大学）；National Center for Supercomputing applications, University of Illinois（伊利诺伊大学超级计算应用国家中心）；University of Illinois（伊利诺伊大学）；Louisiana State University（路易斯安那州立大学）
In the era of gravitational-wave astronomy, general-relativistic simulations of compact objects play a role of paramount importance. These calculations can be performed with the Einstein Toolkit, an open-source and community-supported software for numerical-relativity and relativistic astrophysics. The code comes with multiple solvers for Einstein's equations and for the equations of general-relativistic magneto-hydrodynamics, along with a series of useful diagnostics. However, analyzing the output of the Einstein Toolkit can be a challenging task. Usually, the process involves a series of tec…
We numerically integrate the Einstein's equations for a spatially flat Friedmann-Lemaire-Robertson-Walker (FLRW) background spacetime with a spatial curvature perturbation and evolving primordial tensor perturbations using the Einstein Toolkit. We find that although the primordial tensor perturbation doesn't play an important role in the evolution of the overdensity produced by the scalar perturbation, there is an obvious imprint left by the primordial tensor perturbation on the distribution of the fractional density perturbation in the nonlinear region. This imprint may be a possible probe of…
We perform three-dimensional numerical relativity simulations of homogeneous and inhomogeneous expanding spacetimes, with a view towards quantifying non-linear effects from cosmological inhomogeneities. We demonstrate fourth-order convergence with errors less than one part in 10^6 in evolving a flat, dust Friedmann-Lemaitre-Roberston-Walker (FLRW) spacetime using the Einstein Toolkit within the Cactus framework. We also demonstrate agreement to within one part in 10^3 between the numerical relativity solution and the linear solution for density, velocity and metric perturbations in the Hubble …
We introduce NRPyElliptic, an elliptic solver for numerical relativity (NR) built within the NRPy+ framework. As its first application, NRPyElliptic sets up conformally flat, binary black hole (BBH) puncture initial data (ID) on a single numerical domain, similar to the widely used TwoPunctures code. Unlike TwoPunctures, NRPyElliptic employs a hyperbolic relaxation scheme, whereby arbitrary elliptic PDEs are trivially transformed into a hyperbolic system of PDEs. As consumers of NR ID generally already possess expertise in solving hyperbolic PDEs, they will generally find NRPyElliptic easier t…
GRaM-X: A new GPU-accelerated dynamical spacetime GRMHD code for Exascale computing with the Einstein Toolkit
Anton Pannekoek Institute for Astronomy and GRAPPA, University of Amsterdam（阿姆斯特丹大学安东·潘内库克天文学研究所与GRAPPA）；Institute of High-Energy Physics, University of Amsterdam（阿姆斯特丹大学高能物理研究所）；Center for Computation & Technology, Louisiana State University（路易斯安那州立大学计算与技术中心）；National Center for Supercomputing applications, University of Illinois（伊利诺伊大学国家超级计算应用中心）；Department of Physics, University of Illinois（伊利诺伊大学物理系）；Perimeter Institute for Theoretical Physics（理论物理学珀利默研究所）；Department of Physics and Astronomy, University of Waterloo（滑铁卢大学物理与天文系）；University of Waterloo（滑铁卢大学）；University of Amsterdam（阿姆斯特丹大学）
We use the open source, community-driven, numerical relativity software, the Einstein Toolkit to study the physics of eccentric, spinning, nonprecessing binary black hole mergers with mass-ratios $q=\{2, 4, 6\}$, individual dimensionless spin parameters $χ_{1z}=\pm0.6$, $χ_{2z}=\pm0.3$, that include higher order gravitational wave modes $\ell\leq4$, except for memory modes. Assuming stellar mass binary black hole mergers that may be detectable by the advanced LIGO detectors, we find that including modes up to $\ell=4$ increases the signal-to-noise of compact binaries between $3.5\%$ to $35\%$,…
MHD turbulence is likely to play an important role in several astrophysical scenarios where the magnetic Reynolds is very large. Numerically, these cases can be studied efficiently by means of Large Eddy Simulations, in which the computational resources are used to evolve the system only up to a finite grid size. The resolution is not fine enough to capture all the relevant small-scale physics at play, which is instead effectively modeled by a set of additional terms in the evolution equations, dubbed as sub-grid-scale model. Here we extend such approach, commonly used in non-relativistic/non-…
The tremendous challenge of comparing our theoretical models with the gravitational-wave observations in the new era of multimessenger astronomy requires accurate and fast numerical simulations of complicated physical systems described by the Einstein and the matter equations. These requirements can only be satisfied if the simulations can be parallelized efficiently on a large number of processors and advanced computational strategies are adopted. To achieve this goal we have developed Simflowny, an open platform for scientific dynamical models which automatically generates parallel code for …
Observation and characterisation of gravitational waves from binary black holes requires accurate knowledge of the expected waveforms. The late inspiral and merger phase of the waveform is obtained through direct numerical integration of the full 3-dimensional Einstein equations. The Spectral Einstein Code (SpEC) utilizes a multi-domain pseudo-spectral method tightly adapted to the geometry of the black holes; it is computationally efficient and accurate, but--for high mass-ratios and large spins--sometimes requires manual fine-tuning for the merger-phase of binaries. The Einstein Toolkit (ET)…
Black hole-torus systems from compact binary mergers are possible engines for gamma-ray bursts (GRBs). During the early evolution of the post-merger remnant, the state of the torus is determined by a combination of neutrino cooling and magnetically-driven heating processes, so realistic models must include both effects. In this paper, we study the post-merger evolution of a magnetized black hole-neutron star binary system using the Spectral Einstein Code (SpEC) from an initial post-merger state provided by previous numerical relativity simulations. We use a finite-temperature nuclear equation …
We present GiRaFFE, the first open-source general relativistic force-free electrodynamics (GRFFE) code for dynamical, numerical-relativity generated spacetimes. GiRaFFE adopts the strategy pioneered by McKinney and modified by Paschalidis and Shapiro to convert a GR magnetohydrodynamic (GRMHD) code into a GRFFE code. In short, GiRaFFE exists as a modification of IllinoisGRMHD, a user-friendly, open-source, dynamical-spacetime GRMHD code. Both GiRaFFE and IllinoisGRMHD leverage the Einstein Toolkit's highly-scalable infrastructure to make possible large-scale simulations of magnetized plasmas i…
The applications of numerical relativity to cosmology are on the rise, contributing insight into such cosmological problems as structure formation, primordial phase transitions, gravitational-wave generation, and inflation. In this paper, I present the infrastructure for the computation of inhomogeneous dust cosmologies which was used recently to measure the effect of nonlinear inhomogeneity on the cosmic expansion rate. I illustrate the code's architecture, provide evidence for its correctness in a number of familiar cosmological settings, and evaluate its parallel performance for grids of up…
Gravitational wave surrogate model for spinning, intermediate mass ratio binaries based on perturbation theory and numerical relativity
University of Massachusetts, Dartmouth（达特茅斯马萨诸塞大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Rhode Island（罗德岛大学）；University of California Santa Barbara（加州大学圣塔芭芭拉分校）；University of Virginia（弗吉尼亚大学）；Case Western Reserve University（凯斯西储大学）
The Einstein and Maxwell equations are both systems of hyperbolic equations which need to satisfy a set of elliptic constraints throughout evolution. However, while electrodynamics (EM) and magnetohydrodynamics (MHD) have benefited from a large number of evolution schemes that are able to enforce these constraints and are easily applicable to curvilinear coordinates, unstructured meshes, or N-body simulations, many of these techniques cannot be straightforwardly applied to existing formulations of the Einstein equations. We develop a 3+1 a formulation of the Einstein equations which shows a st…
