An ADM 3+1 formulation for Smooth Lattice General Relativity

A new hybrid scheme for numerical relativity will be presented. The scheme will employ a 3-dimensional spacelike lattice to record the 3-metric while using the standard 3+1 ADM equations to evolve the lattice. Each time step will involve three basic steps. First, the coordinate quantities such as the Riemann and extrinsic curvatures are extracted from the lattice. Second, the 3+1 ADM equations are used to evolve the coordinate data, and finally, the coordinate data is used to update the scalar data on the lattice (such as the leg lengths). The scheme will be presented only for the case of vacuum spacetime though there is no reason why it could not be extended to non-vacuum spacetimes. The scheme allows any choice for the lapse function and shift vectors. An example for the Kasner $T^3$ cosmology will be presented and it will be shown that the method has, for this simple example, zero discretisation error.Comment: 18 pages, plain TeX, 5 epsf figues, gzipped ps file also available at http://newton.maths.monash.edu.au:8000/preprints/3+1-slgr.ps.g

Long term stable integration of a maximally sliced Schwarzschild black hole using a smooth lattice method

We will present results of a numerical integration of a maximally sliced Schwarzschild black hole using a smooth lattice method. The results show no signs of any instability forming during the evolutions to t=1000m. The principle features of our method are i) the use of a lattice to record the geometry, ii) the use of local Riemann normal coordinates to apply the 1+1 ADM equations to the lattice and iii) the use of the Bianchi identities to assist in the computation of the curvatures. No other special techniques are used. The evolution is unconstrained and the ADM equations are used in their standard form.Comment: 47 pages including 26 figures, plain TeX, also available at http://www.maths.monash.edu.au/~leo/preprint

Is the Regge Calculus a consistent approximation to General Relativity?

We will ask the question of whether or not the Regge calculus (and two related simplicial formulations) is a consistent approximation to General Relativity. Our criteria will be based on the behaviour of residual errors in the discrete equations when evaluated on solutions of the Einstein equations. We will show that for generic simplicial lattices the residual errors can not be used to distinguish metrics which are solutions of Einstein's equations from those that are not. We will conclude that either the Regge calculus is an inconsistent approximation to General Relativity or that it is incorrect to use residual errors in the discrete equations as a criteria to judge the discrete equations.Comment: 27 pages, plain TeX, very belated update to match journal articl

High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

Copyright @ 2014 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over inline image for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease

A longitudinal investigation of maternal and child 'anxious cognitions'

Overestimation of threat and underestimation of coping have been frequently reported amongst anxious adults and children. The current study examines the longitudinal relationship between mothers' anxious cognitions and expectations about their child, and children's anxious cognitions. 54 children (aged 10-11 years) and their mothers reported on their interpretation of ambiguous scenarios at two time points. Mothers also reported on their expectations about their child's reaction to ambiguous situations. Significant cross-sectional associations were found between mother and child anticipation of distress. Associations were most consistent between mothers' expectations and children's cognitions. Furthermore, based on regression analyses, mothers' expectations predicted change in children's anxious cognitions over time. Evidence for a reciprocal relationship, that child cognitions predict change in mothers' expectations, was found for girls. The results provide empirical support for potential influences on the development of children's 'anxious cognitive style,' and suggest targets for preventing and reducing maladaptive cognitions in children

Regge Calculus as a Fourth Order Method in Numerical Relativity

The convergence properties of numerical Regge calculus as an approximation to continuum vacuum General Relativity is studied, both analytically and numerically. The Regge equations are evaluated on continuum spacetimes by assigning squared geodesic distances in the continuum manifold to the squared edge lengths in the simplicial manifold. It is found analytically that, individually, the Regge equations converge to zero as the second power of the lattice spacing, but that an average over local Regge equations converges to zero as (at the very least) the third power of the lattice spacing. Numerical studies using analytic solutions to the Einstein equations show that these averages actually converge to zero as the fourth power of the lattice spacing.Comment: 14 pages, LaTeX, 8 figures mailed in separate file or email author directl

Regge calculus and Ashtekar variables

Spacetime discretized in simplexes, as proposed in the pioneer work of Regge, is described in terms of selfdual variables. In particular, we elucidate the "kinematic" structure of the initial value problem, in which 3--space is divided into flat tetrahedra, paying particular attention to the role played by the reality condition for the Ashtekar variables. An attempt is made to write down the vector and scalar constraints of the theory in a simple and potentially useful way.Comment: 10 pages, uses harvmac. DFUPG 83/9

A Smooth Lattice construction of the Oppenheimer-Snyder spacetime

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime. The results are in excellent agreement with theory and numerical results from other authors.Comment: 60 pages, 28 figure

Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology

This is the post-print version of the final published paper that is available from the link below. Copyright @ 2013 Walter de Gruyter GmbH.Non-invasive detection, localization and characterization of an arterial stenosis (a blockage or partial blockage in the artery) continues to be an important problem in medicine. Partial blockage stenoses are known to generate disturbances in blood flow which generate shear waves in the chest cavity. We examine a one-dimensional viscoelastic model that incorporates Kelvin–Voigt damping and internal variables, and develop a proof-of-concept methodology using simulated data. We first develop an estimation procedure for the material parameters. We use this procedure to determine confidence intervals for the estimated parameters, which indicates the efficacy of finding parameter estimates in practice. Confidence intervals are computed using asymptotic error theory as well as bootstrapping. We then develop a model comparison test to be used in determining if a particular data set came from a low input amplitude or a high input amplitude; this we anticipate will aid in determining when stenosis is present. These two thrusts together will serve as the methodological basis for our continuing analysis using experimental data currently being collected.National Institute of Allergy and Infectious Diseases, Air Force Office of Scientific Research, Department of Education, and Engineering and Physical Sciences Research Council