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  • www.monash.edu.au

    Non-standard Computational Methods in Numerical Relativity

    Leo Brewin

    School of Mathematical Sciences

    Monash University

  • Experimental gravity

  • Experimental gravity

  • Experimental gravity

  • Rules of the game

    Numerical relativity

    Construct discrete solutions of Einstein’s equations

    Sounds simple but...

  • Non-standard methods

    Numerical Relativity

    Regge calculus

    Multiquadrics

    Spectral methods

    Smooth lattices

    Tetrad methods Buchman & Bardeen,PhysRevD.67(2003)084017, van Putten, PhysRevD.55(1997)4705

    Discrete differential forms Frauendiener, CQG.23(2006)S369, Richter & Frauendiener, CQG.24(2007)433

    Finite volumes Alic etal,PhysRevD.76(2007)104007

    Finite elements Korobkin etal,CQG.26(2009)145007, Zambusch,CQG.26(2009)175011

  • Multiquadrics

    VOL. 76, NO. 8 7OURNAL OF GEOPHYSICAL RESEARCH MARCH 10, 1•71

    Multiquadric Equations of Topography and Other Irregular Surfaces

    ROLLA•D L.

    Department o] Civil Engineering and Engineering Research Institute Iowa State University, Ames 50010

    A new analytical method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described. The quadric surfaces are located at significant points throughout the region to be mapped. Procedures are given for solving multiquadric equations of topography that are based on coordinate data. Con- toured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived.

    Topography can be represented by various analytical, numerical, and digital methods, in

    addition to the classical contour map. The

    extremes in generalization or detail that result

    from use of these methods are perhaps demon-

    strated best by Lee and Kaula [1967] and by

    Gilbert [1968]. Lee and Kaula described the

    topography of the whole earth in the form of

    thirty-sixth-degree spherical harmonics. Gilbert

    reported the magnetic tape storage of more

    than six million increments of height informa-

    tion in digital form, measured or interpolated

    from one ordinary map sheet. In Lee and Kaula's work we have an ex-

    treme generalization of existing topographic

    information over a wide area by highly analyti-

    cal methods, whereas Gilbert's work is extremely

    detailed but scarcely analytical. As valuable

    as these techniques are in certain cases, they

    are related more to map utilization than to map

    making. Basically, the problem they solve is:

    given continuous topographic information in a

    certain region, reduce it to an equivalent set of

    discrete data, e.g., spherical harmonic coeffi-

    cients or digital terrain increments.

    Other investigators, including myself, are

    concerned with a procedural inverse of the

    above problem, namely: given a set of discrete

    data on a topographic surface, reduce it to a

    satisfactory continuous function representing

    the topographic surface. Practical solutions to

    this problem will tend to eliminate the classical

    Copyright @ 1971 by the American Geophysical Union.

    contour map as the first step in representing terrain information.

    An equation of topography can be evaluated

    digitally or analytically without its having

    been reduced to graphical form. The same equa-

    tion can be treated analytically for the auto-

    matic production of contoured maps. Automatic

    contouring can become a computer-plotter

    problem in analytical geometry, i.e., to deter-

    mine and plot the intercept equations of hori- zontal planes passed through a three-dimen-

    sional equation of topography. This approach could also lead to reconsideration of the need

    for digitized map data. Problems involving map use, such as determining unobstructed

    lines of sight, areas of deftlade, volumes of

    earth, and minimum length of surface curves,

    may involve the more direct application of

    analytical geometry and calculus to the inter-

    relationship of these parameters with a mathe-

    matical surface of topography. For these rea-

    sons, the question of representing a topographic surface in detail by unique equations deserves increased consideration.

    •NUMERICAL SURFACE TEC•NmUES

    Fourier and polynomial series approximations have been applied in recent years to both sur- face and subsurface trend analysis in geological

    mapping. Investigators in this field have been

    Krumbein [1966], Mandelbaum [1963], James

    [1966], and Merriam and Sheath [1966]. There

    has been a natural tendency to apply these

    trend surface techniques to the problem of

    1905

  • Multiquadrics

    Originally used for interpolation on scattered data

    Adapted by Kansa to solve ODEs

    Given compute such that

  • Multiquadrics

    Computers Math. Appfic. Vol. 19, No. 8/9, pp. 127-145, 1990 0097-4943/90 $3.00 + 0.00 Printed in Great Britain Pergamon Press plc

    M U L T I Q U A D R I C S - - A S C A T T E R E D D A T A

    A P P R O X I M A T I O N S C H E M E W I T H A P P L I C A T I O N S T O

    C O M P U T A T I O N A L F L U I D - D Y N A M I C S - - I

    SURFACE APPROXIMATIONS AND PARTIAL DERIVATIVE ESTIMATES

    E. J. KANSA

    Lawrence Livermore National Laboratory, L-200, P.O. Box 808, Livermore, CA 94550, U.S.A.

    A~traet - -We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approxi- mations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy.

    Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results.

    1. BACKGROUND

    The study of arbitrarily shaped curves, surfaces and bodies having arbitrary data orderings has

    immediate application to computational fluid-dynamics. The governing equations not only include

    source terms but gradients, divergences and Laplacians. In addition, many physical processes occur

    over a wide range of length scales. To obtain quantitatively accurate approximations of the physics,

    quantitatively accurate estimates of the spatial variations of such variables are required. In two

    and three dimensions, the range of such quantitatively accurate problems possible on current

    multiprocessing super computers using standard finite difference or finite element codes is limited.

    The question is whether there exist alternative techniques or combinations of techniques which can

    broaden the scope of problems to be solved by permitting steep gradients to be modelled using

    fewer data points. Toward that goal, our study consists of two parts. The first part will investigate

    a new numerical technique of curve, surface and body approximations of exceptional accuracy over

    an arbitrary data arrangement. The second part of this study will use such techniques to improve

    parabolic, hyperbolic or elliptic partial differential equations. We will demonstrate that the study

    of function approximations has a definite advantage to computational methods for partial

    differential equations.

    One very important use of computers is the simulation of multidimensional spatial processes.

    In this paper, we assumed that some finite physical quantity, F, is piecewise continuous in some

    finite domain. In many applications, F is known only at a finite number of locations,

    {xk: k = 1, 2 . . . . . N} where xk = x~ for a univariate problem, and Xk = (x~,yk . . . . )X for the multivariate problem.

    From a finite amount of information regarding F, we seek the best approximation which can

    not only supply accurate estimates of F at arbitrary locations on the domain, but will also provide

    accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain.

    The domain of F will consist of points, {xk }, of arbitrary ordering and sub-clustering. A rectangular

    grid is a very special case of a data ordering.

    Let us assume that an interpolation function, f, approximates F in the sense that

    f(Xk)=F(Xk), k = l , 2 . . . . . N. (1)

    127

  • Example

    solve linear system for

    uniform & random

    cho

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