Non-standard Computational Methods in Numerical Relativity. Leo Brewin School of Mathematical Sciences Monash University.

Transcription

1 Non-standard Computational Methods in Numerical Relativity Leo Brewin School of Mathematical Sciences Monash University

2 Experimental gravity

3 Experimental gravity

4 Experimental gravity

5 Numerical relativity Rules of the game Construct discrete solutions of Einstein s equations Sounds simple but...

6 Numerical Relativity Non-standard methods Multiquadrics Spectral methods Regge calculus Smooth lattices Discrete differential forms Frauendiener, CQG.23(2006)S369, Richter & Frauendiener, CQG.24(2007)433 Tetrad methods Finite elements Finite volumes Buchman & Bardeen,PhysRevD.67(2003)084017, van Putten, PhysRevD.55(1997)4705 Korobkin etal,cqg.26(2009)145007, Zambusch,CQG.26(2009) Alic etal,physrevd.76(2007)104007

7 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 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. Contoured multiquadric surfaces are compared with topography and other irregular surfaces from which the multiquadric equation was derived.

8 Multiquadrics Originally used for interpolation on scattered data Given compute such that Adapted by Kansa to solve ODEs

9 Multiquadrics Computers Math. Appfic. Vol. 19, No. 8/9, pp , /90 $ Printed in Great Britain Pergamon Press plc MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS--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 approximations. 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.

10 Example choose uniform & random solve linear system for reconstruct interpolant

11 Example linear system for Bad news is dense is almost singular, condition number typically Must use Singular Value Decomposition, expensive Good news Convergence is very rapid, approx. exponential

12 Bowen-York initial data

13 Bowen-York initial data

14 Bowen-York initial data

15 Pros and cons Free to place nodes wherever we like. Exponential convergence. Very accurate for very steep functions. Must solve exceedingly ill-conditioned system. Requires care for asymptoticly flat geometries. Only two papers with results, volunteers most welcome.

16 Spectral methods Living Rev. Relativity, 12, (2009), 1 Spectral Methods for Numerical Relativity Philippe Grandclément Laboratoire Univers et Théories UMR 8102 du C.N.R.S., Observatoire de Paris F Meudon Cedex, France Philippe.Grandclement@obspm.fr Jérôme Novak Laboratoire Univers et Théories UMR 8102 du C.N.R.S., Observatoire de Paris F Meudon Cedex, France Jerome.Novak@obspm.fr

17 A gentle introduction

18 Spectral interpolants is a function is a polynomial approximation of degree to

19 Convergence N=4 N= u = cos 3 (!x/2) - (x+1) 3 /8 P u I u -1 u = cos 3 (!x/2) - (x+1) 3 /8 P u I u x x

20 Exponential convergence 1e+00 1e-03 max! I N u - u 1e-06 1e-09 1e-12 1e Number of coefficients

21 Gibbs phenomena x

22 Representations can be represented by either or for algebraic operators use for differential operators use

23 Differential equations solve for then recover

24 Spectral methods in Numerical Relativity PHYSICAL REVIEW D, VOLUME 62, Black hole evolution by spectral methods Lawrence E. Kidder, Mark A. Scheel, and Saul A. Teukolsky Center for Radiophysics and Space Research, Cornell University, Ithaca, New York Eric D. Carlson and Gregory B. Cook Department of Physics, Wake Forest University, Winston-Salem, North Carolina Received 15 May 2000; published 26 September 2000 Current methods of evolving a spacetime containing one or more black holes are plagued by instabilities that prohibit long-term evolution. Some of these instabilities may be due to the numerical method used, traditionally finite differencing. In this paper, we explore the use of a pseudospectral collocation PSC method for the evolution of a spherically symmetric black hole spacetime in one dimension using a hyperbolic formulation of Einstein s equations. We demonstrate that our PSC method is able to evolve a spherically symmetric black hole spacetime forever without enforcing constraints, even if we add dynamics via a Klein-Gordon scalar field. We find that, in contrast with finite-differencing methods, black hole excision is a trivial operation using PSC applied to a hyperbolic formulation of Einstein s equations. We discuss the extension of this method to three spatial dimensions.

25 Stable evolutions in 1+1 d FIG. 1. Long-term stability of the evolution of Kerr-Schild initial data, run 1 from Table I. Plotted is the l 2 norm of the Hamiltonian constraint 2.20a in units of M 2 as a function of time for several spatial resolutions. The number of spectral coefficients N r for each plot, starting at the top, is 12, 16, 20, 24, 27, 32, 36, 40, 45, 48, 54, and 60.

26 Pros and cons Exponential convergence. Superb results for little effort. No dissipation. Gibbs phenomena. Discrete equations are fully coupled. Lack of dissipation allows high frequency errors to remain. Can not freely choose location of nodes.

27 Regge calculus

28 Regge calculus

29 2d example

30 2d example Metric recorded by table of leg-lengths Topology recorded by connection matrix

31 2d example Metric is piecewise flat Curvature is a distribution Defect angle at each vertex

32 2d example

33 2d example

34 2d example

35 2d example

36 2d example

37 3d example

38 3d example

39 3d example

40 3d example

41 3d example

42 4d example

43 Field equations

44 Some questions Can smooth metrics be accurately approximated by Regge lattices? Do the Regge equations reduce to the Einstein equations in some suitable limit? Do solutions of the Regge equations converge to solutions of the Einstein equations?

45 Some answers Continuum Regge Yes Allendorfer, Weyl Yes, as an average Cheeger, Muller, Schrader Einstein solutions? Regge solutions

46 Applications Kasner cosmologies Quantum gravity Brill wave initial data Misner initial data FRW cosmologies

47 Kasner cosmology Class. Quantum Grav. 15 (1998) Printed in the UK PII: S (98) A fully (3 + 1)-dimensional Regge calculus model of the Kasner cosmology Adrian P Gentle and Warner A Miller Theoretical Division (T-6, MS B288), Los Alamos National Laboratory, Los Alamos, NM 87545, USA Received 5 June 1997, in final form 23 October 1997 Abstract. We describe the first discrete-time four-dimensional numerical application of Regge calculus. The spacetime is represented as a complex of four-dimensional simplices, and the geometry interior to each 4-simplex is flat Minkowski spacetime. This simplicial spacetime is constructed so as to be foliated with a one-parameter family of spacelike hypersurfaces built from tetrahedra. We implement a novel 2-surface initial-data prescription for Regge calculus, and provide the first fully four-dimensional application of an implicit decoupled evolution scheme (the Sorkin evolution scheme ). We benchmark this code on the Kasner cosmology a cosmology which embodies generic features of the collapse of many cosmological models. We (i) reproduce the continuum solution with a fractional error in the 3-volume of 10 5 after evolution steps; (ii) demonstrate stable evolution; (iii) preserve the standard deviation of spatial homogeneity to less than and (iv) explicitly display the existence of diffeomorphism freedom in Regge calculus. We also present the second-order convergence properties of the solution to the continuum.

48 Kasner cosmology Vacuum, homogenous and isotropic

49 Kasner lattice

50 Kasner lattice

51 Kasner lattice

52 Kasner lattice

53 Sorkin evolution

54 Kasner cosmology

55 Brill wave initial data Class. Quantum Grav. 16 (1999) Printed in the UK PII: S (99) Simplicial Brill wave initial data Adrian P Gentle Department of Mathematics and Statistics, Monash University, Clayton, Victoria 3168, Australia and Theoretical Division (T-6, MS B288), Los Alamos National Laboratory, Los Alamos, NM 87545, USA adrian@newton.maths.monash.edu.au Received 12 January 1999 Abstract. Regge calculus is used to construct initial data for vacuum axisymmetric Brill waves at a moment of time symmetry. We argue that only a tetrahedral lattice can successfully reproduce the continuum solution, and develop a simplicial axisymmetric lattice based on the coordinate structure of the continuum metric. This is used to construct initial data for Brill waves in an otherwise flat spacetime, and for the distorted black hole spacetime of Bernstein. These initial data sets are shown to be second-order accurate approximations to the corresponding continuum solutions.

56 Brill wave initial data Time symmetric initial data Given compute One equation per point One equation per vertex

57 Brill wave initial data Figure 3. A section of the axisymmetric lattice used by Dubal [14]. The relation to the global polar coordinate system is shown, together with the angles used to specify the extra degree of freedom. Figure 4. The rectangular prism shown in figure 3 subdivided into six tetrahedra. This involves adding a diagonal brace to each face of the prism, together with a body diagonal.

58 Brill wave conformal factor Figure 1. The conformal factor ψ for Brill wave initial data, calculated on a grid using a centred finite-difference approximation to equation (7). The Eppley form of q(ρ, z) (equation (8)) was used, with wave amplitude a = 10, and the outer boundaries were placed at ρ = 20 and z = 20. Figure 5. The conformal factor ψ for a Brill wave of amplitude a = 10 calculated using the simplicial Regge lattice. The agreement between this and the continuum solution in figure 1 is excellent. The calculation was performed using a lattice consisting of vertices, with the outer boundaries at ρ = 20 and z = 20.

59 Brill wave lattice convergence

60 Pros and cons Coordinate free. Clear separation between topology and metric. Local computations are trivial. Metric not smooth. Must treat curvature as a distribution. Analysis is hard, very few theorems. Convergence is unclear.

61 Smooth lattice GR

62 Smooth lattice GR Originally called Chicken wire relativity

63 2d example

64 2d example Metric recorded by table of leg-lengths Topology recorded by connection matrix

65 2d example Metric is smooth Legs are geodesic segments Curvature is a point function

66 Riemann normal coordinates etc.

67 Geodesic leg lengths

68 General relativity Given solve for and Then impose the vacuum field equations is now an algebraic system for the Solve the equations. Et voila -- a numerical spacetime!

69 ADM equations on a lattice With zero shift and unit lapse Applied to the lattice

70 A quick derivation Differentiate twice and retain leading order terms Alternatively, can use 2nd variation of arc-length

71 Examples 1+1 evolution Maximal & geodesic slicing of Schwarzschild Oppenheimer-Snyder dust collapse (with Jules Kajtar) 3+1 evolution Flat Kasner cosmologies Teukolsky spacetime (in progress)

72 Maximally sliced Schwarzschild INSTITUTE OF PHYSICS PUBLISHING Class. Quantum Grav. 19 (2002) CLASSICAL AND QUANTUM GRAVITY PII: S (02) Long term stable integration of a maximally sliced Schwarzschild black hole using a smooth lattice method Leo Brewin Department of Mathematics & Statistics, Monash University, Clayton, Victoria 3800, Australia Received 27 July 2001 Published 14 January 2002 Online at stacks.iop.org/cqg/19/429 Abstract 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 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.

73 Schwarzschild

74 The lattice Spherical symmetry, so only need one ladder Edges of ladder are radial geodesics Evolve the rungs and radial segments

75 Evolution equations Note, now using 3d Riemann curvatures

76 Curvature equations Geodesic deviation Bianchi identity

77 Collapse of the lapse t=10m-100m t=100m-1000m 1.00e e e e e e-01 N N 4.00e e e e e Radial proper distance e Radial proper distance e e e e-50 Central lapse N 1.00e e-15 Central lapse N 1.00e e e e e Maximal time τ e Maximal time τ

78 Oppenheimer-Snyder dust collapse PHYSICAL REVIEW D 80, (2009) Smooth lattice construction of the Oppenheimer-Snyder spacetime Leo Brewin* and Jules Kajtar School of Mathematical Sciences Monash University, 3800 Australia (Received 23 May 2009; published 5 November 2009) 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.

79 Lattice FRW dust interior Schwarzschild exterior

80 Evolution of the curvature t=0-60m

81 Evolution of the curvature t=0-200m

82 Pros and cons All the pros of Regge calculus plus... Metric is differentiable. Curvature is a pointwise function. Can use all the usual mathematical tools. Lacks empirical support, but results are promising. Must solve a coupled system to compute curvatures. Coordinates not known apriori. How do we impose boundary conditions? No non-symmetric 3+1 results...but stay tuned to gr-qc.

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