Black Hole-Neutron Star Binaries in General Relativity: Quasiequilibrium Formulation

We present a new numerical method for the construction of quasiequilibrium models of black hole-neutron star binaries. We solve the constraint equations of general relativity, decomposed in the conformal thin-sandwich formalism, together with the Euler equation for the neutron star matter. We take the system to be stationary in a corotating frame and thereby assume the presence of a helical Killing vector. We solve these coupled equations in the background metric of a Kerr-Schild black hole, which accounts for the neutron star's black hole companion. In this paper we adopt a polytropic equation of state for the neutron star matter and assume large black hole--to--neutron star mass ratios. These simplifications allow us to focus on the construction of quasiequilibrium neutron star models in the presence of strong-field, black hole companions. We summarize the results of several code tests, compare with Newtonian models, and locate the onset of tidal disruption in a fully relativistic framework.Comment: 17 pages, 7 figures; added discussion, tables; PRD in pres

A spherical shell numerical dynamo benchmark with pseudo vacuum magnetic boundary conditions

It is frequently considered that many planetary magnetic fields originate as a result of convection within planetary cores. Buoyancy forces responsible for driving the convection generate a fluid flow that is able to induce magnetic fields; numerous sophisticated computer codes are able to simulate the dynamic behaviour of such systems. This paper reports the results of a community activity aimed at comparing numerical results of several different types of computer codes that are capable of solving the equations of momentum transfer, magnetic field generation and heat transfer in the setting of a spherical shell, namely a sphere containing an inner core. The electrically conducting fluid is incompressible and rapidly rotating and the forcing of the flow is thermal convection under the Boussinesq approximation. We follow the original specifications and results reported in Harder & Hansen to construct a specific benchmark in which the boundaries of the fluid are taken to be impenetrable, non-slip and isothermal, with the added boundary condition for the magnetic field <b>B</b> that the field must be entirely radial there; this type of boundary condition for <b>B</b> is frequently referred to as ‘pseudo-vacuum’. This latter condition should be compared with the more frequently used insulating boundary condition. This benchmark is so-defined in order that computer codes based on local methods, such as finite element, finite volume or finite differences, can handle the boundary condition with ease. The defined benchmark, governed by specific choices of the Roberts, magnetic Rossby, Rayleigh and Ekman numbers, possesses a simple solution that is steady in an azimuthally drifting frame of reference, thus allowing easy comparison among results. Results from a variety of types of code are reported, including codes that are fully spectral (based on spherical harmonic expansions in angular coordinates and polynomial expansions in radius), mixed spectral and finite difference, finite volume, finite element and also a mixed Fourier-finite element code. There is good agreement among codes

How universal is the fractional-quantum-Hall edge Luttinger liquid?

This article reports on our microscopic investigations of the edge of the fractional quantum Hall state at filling factor $\nu=1/3$. We show that the interaction dependence of the wave function is well described in an approximation that includes mixing with higher composite-fermion Landau levels in the lowest order. We then proceed to calculate the equal time edge Green function, which provides evidence that the Luttinger exponent characterizing the decay of the Green function at long distances is interaction dependent. The relevance of this result to tunneling experiments is discussed.Comment: 5 page

Pseudospectral Calculation of Helium Wave Functions, Expectation Values, and Oscillator Strength

The pseudospectral method is a powerful tool for finding highly precise solutions of Schr\"{o}dinger's equation for few-electron problems. We extend the method's scope to wave functions with non-zero angular momentum and test it on several challenging problems. One group of tests involves the determination of the nonrelativistic electric dipole oscillator strength for the helium $1^1$S $\to 2^1$P transition. The result achieved, $0.27616499(27)$, is comparable to the best in the literature. Another group of test applications is comprised of well-studied leading order finite nuclear mass and relativistic corrections for the helium ground state. A straightforward computation reaches near state-of-the-art accuracy without requiring the implementation of any special-purpose numerics. All the relevant quantities tested in this paper -- energy eigenvalues, S-state expectation values and bound-bound dipole transitions for S and P states -- converge exponentially with increasing resolution and do so at roughly the same rate. Each individual calculation samples and weights the configuration space wave function uniquely but all behave in a qualitatively similar manner. Quantum mechanical matrix elements are directly and reliably calculable with pseudospectral methods. The technical discussion includes a prescription for choosing coordinates and subdomains to achieve exponential convergence when two-particle Coulomb singularities are present. The prescription does not account for the wave function's non-analytic behavior near the three-particle coalescence which should eventually hinder the rate of the convergence. Nonetheless the effect is small in the sense that ignoring the higher-order coalescence does not appear to affect adversely the accuracy of any of the quantities reported nor the rate at which errors diminish.Comment: 24 pages, 12 figures, 6 tables. To be submitted to Physical Review A. LANL identifier 'LA-UR-11-10986

Extended Lifetime in Computational Evolution of Isolated Black Holes

Solving the 4-d Einstein equations as evolution in time requires solving equations of two types: the four elliptic initial data (constraint) equations, followed by the six second order evolution equations. Analytically the constraint equations remain solved under the action of the evolution, and one approach is to simply monitor them ({\it unconstrained} evolution). The problem of the 3-d computational simulation of even a single isolated vacuum black hole has proven to be remarkably difficult. Recently, we have become aware of two publications that describe very long term evolution, at least for single isolated black holes. An essential feature in each of these results is {\it constraint subtraction}. Additionally, each of these approaches is based on what we call "modern," hyperbolic formulations of the Einstein equations. It is generally assumed, based on computational experience, that the use of such modern formulations is essential for long-term black hole stability. We report here on comparable lifetime results based on the much simpler ("traditional") $\dot g$ - $\dot K$ formulation. We have also carried out a series of {\it constrained} 3-d evolutions of single isolated black holes. We find that constraint solution can produce substantially stabilized long-term single hole evolutions. However, we have found that for large domains, neither constraint-subtracted nor constrained $\dot g$ - $\dot K$ evolutions carried out in Cartesian coordinates admit arbitrarily long-lived simulations. The failure appears to arise from features at the inner excision boundary; the behavior does generally improve with resolution.Comment: 20 pages, 6 figure

Finding apparent horizons and other two-surfaces of constant expansion

Apparent horizons are structures of spacelike hypersurfaces that can be determined locally in time. Closed surfaces of constant expansion (CE surfaces) are a generalisation of apparent horizons. I present an efficient method for locating CE surfaces. This method uses an explicit representation of the surface, allowing for arbitrary resolutions and, in principle, shapes. The CE surface equation is then solved as a nonlinear elliptic equation. It is reasonable to assume that CE surfaces foliate a spacelike hypersurface outside of some interior region, thus defining an invariant (but still slicing-dependent) radial coordinate. This can be used to determine gauge modes and to compare time evolutions with different gauge conditions. CE surfaces also provide an efficient way to find new apparent horizons as they appear e.g. in binary black hole simulations.Comment: 21 pages, 8 figures; two references adde

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519. Author's addresses: E. Romero, Institut I3M, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain), and J. E. Roman, Departament de Sistemes Informatics i Computacio, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain; email: [email protected] Alcalde, E.; Román Moltó, JE. (2014). A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Transactions on Mathematical Software. 40(2):13:01-13:29. https://doi.org/10.1145/2543696S13:0113:29402P. Arbenz, M. Becka, R. Geus, U. Hetmaniuk, and T. Mengotti. 2006. On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Comput. 32, 2, 157--165.Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds. 2000. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, PA.C. G. Baker, U. L. Hetmaniuk, R. B. Lehoucq, and H. K. Thornquist. 2009. Anasazi software for the numerical solution of large-scale eigenvalue problems. ACM Trans. Math. Softw. 36, 3, 13:1--13:23.S. Balay, J. Brown, K. Buschelman, V. Eijkhout, W. Gropp, D. Kaushik, M. Knepley, L. C. McInnes, B. Smith, and H. Zhang. 2011. PETSc users manual. Tech. Rep. ANL-95/11-Revision 3.2, Argonne National Laboratory.S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith. 1997. Efficient management of parallelism in object oriented numerical software libraries. In Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, Eds., Birkhaüser, 163--202.M. A. Brebner and J. Grad. 1982. Eigenvalues of Ax =λ Bx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process. Linear Algebra Appl. 43, 99--118.C. Campos, J. E. Roman, E. Romero, and A. Tomas. 2011. SLEPc users manual. Tech. Rep. DSICII/24/02 - Revision 3.2, D. Sistemes Informàtics i Computació, Universitat Politècnica de València. http://www.grycap.upv.es/slepc.T. Dannert and F. Jenko. 2005. Gyrokinetic simulation of collisionless trapped-electronmode turbulence. Phys. Plasmas 12, 7, 072309.E. R. Davidson. 1975. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17, 1, 87--94.T. A. Davis and Y. Hu. 2011. The University of Florida Sparse Matrix Collection. ACM Trans. Math. Softw. 38, 1, 1:1--1:25.H. C. Elman, A. Ramage, and D. J. Silvester. 2007. Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33, 2. Article 14.T. Ericsson and A. Ruhe. 1980. The spectral transformation Lanczos method for the numerical solution of large sparse generalized symmetric eigenvalue problems. Math. Comp. 35, 152, 1251--1268.M. Ferronato, C. Janna, and G. Pini. 2012. Efficient parallel solution to large-size sparse eigenproblems with block FSAI preconditioning. Numer. Linear Algebra Appl. 19, 5, 797--815.D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst. 1998. Jacobi--Davidson style QR and QZ algorithms for the reduction of matrix pencils. SIAM J. Sci. Comput. 20, 1, 94--125.M. A. Freitag and A. Spence. 2007. Convergence theory for inexact inverse iteration applied to the generalised nonsymmetric eigenproblem. Electron. Trans. Numer. Anal. 28, 40--64.M. Genseberger. 2010. Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi-Davidson method for large scale eigenvalue problems. App. Numer. Math. 60, 11, 1083--1099.V. Hernandez, J. E. Roman, and A. Tomas. 2007. Parallel Arnoldi eigensolvers with enhanced scalability via global communications rearrangement. Parallel Comput. 33, 7--8, 521--540.V. Hernandez, J. E. Roman, and V. Vidal. 2005. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31, 3, 351--362.V. Heuveline, B. Philippe, and M. Sadkane. 1997. Parallel computation of spectral portrait of large matrices by Davidson type methods. Numer. Algor. 16, 1, 55--75.M. E. Hochstenbach. 2005a. Generalizations of harmonic and refined Rayleigh-Ritz. Electron. Trans. Numer. Anal. 20, 235--252.M. E. Hochstenbach. 2005b. Variations on harmonic Rayleigh--Ritz for standard and generalized eigenproblems. Preprint, Department of Mathematics, Case Western Reserve University.M. E. Hochstenbach and Y. Notay. 2006. The Jacobi--Davidson method. GAMM Mitt. 29, 2, 368--382.F.-N. Hwang, Z.-H. Wei, T.-M. Huang, and W. Wang. 2010. A parallel additive Schwarz preconditioned Jacobi-Davidson algorithm for polynomial eigenvalue problems in quantum dot simulation. J. Comput. Phys. 229, 8, 2932--2947.A. V. Knyazev. 2001. Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 2, 517--541.A. V. Knyazev, M. E. Argentati, I. Lashuk, and E. E. Ovtchinnikov. 2007. Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in HYPRE and PETSc. SIAM J. Sci. Comput. 29, 5, 2224--2239.J. Kopal, M. Rozložník, M. Tuma, and A. Smoktunowicz. 2012. Rounding error analysis of orthogonalization with a non-standard inner product. Numer. Math. 52, 4, 1035--1058.D. Kressner. 2006. Block algorithms for reordering standard and generalized Schur forms. ACM Trans. Math. Softw. 32, 4, 521--532.R. B. Lehoucq, D. C. Sorensen, and C. Yang. 1998. ARPACK Users' Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA.Z. Li, Y. Saad, and M. Sosonkina. 2003. pARMS: a parallel version of the algebraic recursive multilevel solver. Numer. Linear Algebra Appl. 10, 5--6, 485--509.J. R. McCombs and A. Stathopoulos. 2006. Iterative validation of eigensolvers: a scheme for improving the reliability of Hermitian eigenvalue solvers. SIAM J. Sci. Comput. 28, 6, 2337--2358.F. Merz, C. Kowitz, E. Romero, J. E. Roman, and F. Jenko. 2012. Multi-dimensional gyrokinetic parameter studies based on eigenvalues computations. Comput. Phys. Commun. 183, 4, 922--930.R. B. Morgan. 1990. Davidson's method and preconditioning for generalized eigenvalue problems. J. Comput. Phys. 89, 241--245.R. B. Morgan. 1991. Computing interior eigenvalues of large matrices. Linear Algebra Appl. 154--156, 289--309.R. B. Morgan and D. S. Scott. 1986. Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices. SIAM J. Sci. Statist. Comput. 7, 3, 817--825.R. Natarajan and D. Vanderbilt. 1989. A new iterative scheme for obtaining eigenvectors of large, real-symmetric matrices. J. Comput. Phys. 82, 1, 218--228.M. Nool and A. van der Ploeg. 2000. A parallel Jacobi--Davidson-type method for solving large generalized eigenvalue problems in magnetohydrodynamics. SIAM J. Sci. Comput. 22, 1, 95--112.J. Olsen, P. Jørgensen, and J. Simons. 1990. Passing the one-billion limit in full configuration-interaction (FCI) calculations. Chem. Phys. Lett. 169, 6, 463--472.C. C. Paige, B. N. Parlett, and H. A. van der Vorst. 1995. Approximate solutions and eigenvalue bounds from Krylov subspaces. Numer. Linear Algebra Appl. 2, 2, 115--133.E. Romero and J. E. Roman. 2011. Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi--Davidson eigensolver. Concur. Comput.: Pract. Exp. 23, 17, 2179--2191.Y. Saad. 1993. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 2, 461--469.G. L. G. Sleijpen, A. G. L. Booten, D. R. Fokkema, and H. A. van der Vorst. 1996. Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36, 3, 595--633.G. L. G. Sleijpen and H. A. van der Vorst. 1996. A Jacobi--Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 2, 401--425.G. L. G. Sleijpen and H. A. van der Vorst. 2000. A Jacobi--Davidson iteration method for linear eigenvalue problems. SIAM Rev. 42, 2, 267--293.G. L. G. Sleijpen, H. A. van der Vorst, and E. Meijerink. 1998. Efficient expansion of subspaces in the Jacobi--Davidson method for standard and generalized eigenproblems. Electron. Trans. Numer. Anal. 7, 75--89.A. Stathopoulos. 2007. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part I: Seeking one eigenvalue. SIAM J. Sci. Comput. 29, 2, 481--514.A. Stathopoulos and J. R. McCombs. 2007. Nearly optimal preconditioned methods for Hermitian eigenproblems under limited memory. Part II: Seeking many eigenvalues. SIAM J. Sci. Comput. 29, 5, 2162--2188.A. Stathopoulos and J. R. McCombs. 2010. PRIMME: PReconditioned Iterative MultiMethod Eigensolver: Methods and software description. ACM Trans. Math. Softw. 37, 2, 21:1--21:30.A. Stathopoulos and Y. Saad. 1998. Restarting techniques for the (Jacobi-)Davidson symmetric eigenvalue methods. Electron. Trans. Numer. Anal. 7, 163--181.A. Stathopoulos, Y. Saad, and C. F. Fischer. 1995. Robust preconditioning of large, sparse, symmetric eigenvalue problems. J. Comput. Appl. Math. 64, 3, 197--215.A. Stathopoulos, Y. Saad, and K. Wu. 1998. Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods. SIAM J. Sci. Comput. 19, 1, 227--245.G. W. Stewart. 2001. Matrix Algorithms. Volume II: Eigensystems. SIAM, Philadelphia, PA.H. A. van der Vorst. 2002. Computational methods for large eigenvalue problems. In Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, Eds., Vol. VIII, Elsevier, 3--179.H. A. van der Vorst. 2004. Modern methods for the iterative computation of eigenpairs of matrices of high dimension. Z. Angew. Math. Mech. 84, 7, 444--451.T. van Noorden and J. Rommes 2007. Computing a partial generalized real Schur form using the Jacobi--Davidson method. Numer. Linear Algebra Appl. 14, 3, 197--215.T. D. Young, E. Romero, and J. E. Roman. 2013. Parallel finite element density functional computations exploiting grid refinement and subspace recycling. Comput. Phys. Commun. 184, 1, 66--72

The expression, localisation and interactome of pigeon CRY2

Cryptochromes (CRY) are highly conserved signalling molecules that regulate circadian rhythms and are candidate radical pair based magnetoreceptors. Birds have at least four cryptochromes (CRY1a, CRY1b, CRY2, and CRY4), but few studies have interrogated their function. Here we investigate the expression, localisation and interactome of clCRY2 in the pigeon retina. We report that clCRY2 has two distinct transcript variants, clCRY2a, and a previously unreported splice isoform, clCRY2b which is larger in size. We show that clCRY2a mRNA is expressed in all retinal layers and clCRY2b is enriched in the inner and outer nuclear layer. To define the localisation and interaction network of clCRY2 we generated and validated a monoclonal antibody that detects both clCRY2 isoforms. Immunohistochemical studies revealed that clCRY2a/b is present in all retinal layers and is enriched in the outer limiting membrane and outer plexiform layer. Proteomic analysis showed clCRY2a/b interacts with typical circadian molecules (PER2, CLOCK, ARTNL), cell junction proteins (CTNNA1, CTNNA2) and components associated with the microtubule motor dynein (DYNC1LI2, DCTN1, DCTN2, DCTN3) within the retina. Collectively these data show that clCRY2 is a component of the avian circadian clock and unexpectedly associates with the microtubule cytoskeleton

Hermes : global plasma edge fluid turbulence simulations

The transport of heat and particles in the relatively collisional edge regions of magnetically confined plasmas is a scientifically challenging and technologically important problem. Understanding and predicting this transport requires the self-consistent evolution of plasma fluctuations, global profiles and flows, but the numerical tools capable of doing this in realistic (diverted) geometry are only now being developed. Here a 5-field reduced 2-fluid plasma model for the study of instabilities and turbulence in magnetised plasmas is presented, built on the BOUT++ framework. This cold ion model allows the evolution of global profiles, electric fields and flows on transport timescales, with flux-driven cross-field transport determined self-consistently by electromagnetic turbulence. Developments in the model formulation and numerical implementation are described, and simulations are performed in poloidally limited and diverted tokamak configurations

A Multiscale Model of Partial Melts 2: Numerical Results

In a companion paper, equations for partially molten media were derived using two-scale homogenization theory. One advantage of homogenization is that material properties, such as permeability and viscosity, readily emerge. A caveat is that the dependence of these parameters upon the microstructure is not self-evident. In particular, one seeks to relate them to the porosity. In this paper, we numerically solve ensembles of the cell problems from which these quantities emerge. Using this data, we estimate relationships between the parameters and the porosity. In particular, the bulk viscosity appears to be inversely proportional to the porosity. Finally, we synthesize these numerical estimates with the models. Our hybrid numerical--analytical model predicts that the compaction length vanishes with porosity.Comment: 50 pages, submitted to JGR Solid Eart