New Numerical Code for Black Hole Initial Data

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1 Marshall University Marshall Digital Scholar Physics Faculty Research Physics Summer New Numerical Code for Black Hole Initial Data Maria Babiuc-Hamilton Marshall University, Follow this and additional works at: Part of the Physics Commons Recommended Citation Hamilton, M. & Winicour, J. (2017, June). New Numerical Code for Black Hole Initial Data. Slides presented at the 20th Eastern Gravity Meeting. Penn State, PA. This Presentation is brought to you for free and open access by the Physics at Marshall Digital Scholar. It has been accepted for inclusion in Physics Faculty Research by an authorized administrator of Marshall Digital Scholar. For more information, please contact

2 New Numerical Code for Black Hole Initial Data HyperSolID Maria Babiuc Hamilton and Jeff Winicour 20th Eastern Gravity Meeting Penn State, University Park, PA June 9, 2017

3 1 Introduction Problems Approaches HyperSolID 2 Formulation Equations Variables Calculations Separation Transformation Projection 3 Implementation Algorithm Numerics Testing 4 Timeline Outline

4 Problems Main Problems There are no no exact solutions of Einstein Equations that describes a bound system radiating gravitational waves. One needs to resort to numerical simulations, or analytical approximation methods. Current methods to constrained initial data exhibit junk radiation and ambiguities about constrained and free data. Possible Solutions It was mathematically proved that given the correct initial data, Einstein equation will yield the expected solution. New formulations to calculate the constrains ensuring that the numerical system is well-behaved (hyperbolic equations).

5 Standard Approach Approaches The constraints are formulated as elliptic equations (ex: the conformal thin sandwich method, the gluing technique, etc.) D j K j i D i K j j = 0, (3) R + ( K j j) 2 Kij K ij = 0, New Approach The 3D initial surfaces are further foliated into 2D surfaces, where the constraints form an algebraic-hyperbolic system: Lˆn K ˆD l k l + F K = 0, Lˆn k i + K 1 (κ ˆD i K 2k l ˆD i k l ) + F ki = 0 κ = (2K) 1 ( 2k l k l 1 2 K2 κ 0 )

6 HyperSolID Hyperbolic Solver for Initial Data Description 1 General 4D metric is given on the initial time slice 2 Free 3D initial data is constructed from the metric 3 Constrained 2D initial data given on the initial sphere The constrained data is updated by solving the hyperbolic-algebraic system.

7 1 Introduction Problems Approaches HyperSolID 2 Formulation Equations Variables Calculations Separation Transformation Projection 3 Implementation Algorithm Numerics Testing 4 Timeline Outline

8 Equations Evolution Equations ρ U = 1 2ÑðU + 1 2ÑðU + 1 Nd 1 [ 2 a(ðv + ðv) bðv bðv ] F U ρ v = 2Ñðv + 2Ñðv NU NU 1 { wðu d 1 [(av bv)ðv + (av bv)ðv] } f v Algebraic Equation w = (2U) 1 [ d 1 (2avv bv 2 bv 2 ) 1 2 U2 κ 0 ]

9 Known Variables The known terms that enter in these formulas are: N, (Ñ, Ñ), (ð N, ð N), (ðñ, ðñ, ðñ, ðñ), Variables a, (b, b), d, (ða, ða), (ðb, ðb, ðb, ðb), (A, A), (B, B), (C, C) κ 0, K, K, ( K, K), ( K, K), K, ðκ 0, (ð K, ð K)(ð K, ð K). Unknown Variables The unknown variables are: (U, v, w) and their derivatives:. (ðu, ðu), (ðv, ðv, ðv, ðv) The angular derivatives are the Newman-Penrose (ð, ð) operators

10 Start-up data Calculations We start with a general 4D metric, (g ij, t g ij ), in Cartesian coordinates (x, y, z). With this data we calculate: g ij, n i, h ij, h ij = g ij + n i n j, ( k h ij, k h ij, m k h ij ) 3 Γ i jk,3 R i jlk, 3 R jk, 3 R, K ij, K i i Radial Foliation We choose a simple spherical foliation ρ = r, where r = δ ij x i x j. For this foliation we calculate the quantities : N = (h ij i ρ j ρ) 1/2, N i, n i, γ ij = h ij n i n j, k γ ij, ρ γ ij, Γ i jk, K ij, K.

11 Separation Extrinsic Curvature The extrinsic curvature is decomposed in: K ij = κˆn i ˆn j + 2ˆn (i k j) + K ij, where K ij = γ k i γ l j K kl Constrained Data κ = n i n j K ij, k i = γ j i nk K jk, K = γ ij K ij Free Data K ij = K ij 1 2 γ ijk, κ 0 = 3 R K kl K kl.

12 Transformation Change of Coordinates From Cartesian (x, y, z) to stereographic coordinates (r, q, p) x = Ωrq, y = ±Ωrp, z = 1 2 Ωr( 1+q2 +p 2 ), Ω = (1+q 2 +p 2 ) 1. The coordinates of the unit sphere metric q ab = Ω 2 δ ab, are: q a = Ω 1 (1, i), q a = Ω(1, i), Transformed variables We transform to the stereographic coordinates the variables: (γ ij, K ij, K ij ), ( N i, k i ), ( N, K, K, κ 0, κ)

13 Projection Calculation of the stereographic projections Metric terms a = 1 2 qi q j γ ij, b = 1 2 qi q j γ ij, d = a 2 bb, A = d 1 [ a(2ða ðb) bðb ], B = d 1 [ aðb bðb ] C = d 1 [ aðb b(2ða ðb) ], Ñ = q i N i. Given terms K = q i q j Kij, K = q i q j Kij, K = q i q j K ij, K = q i q j K ij. Updated terms U = K, v = q i k i, w = κ.

14 1 Introduction Problems Approaches HyperSolID 2 Formulation Equations Variables Calculations Separation Transformation Projection 3 Implementation Algorithm Numerics Testing 4 Timeline Outline

15 Algorithm Algebra 1 Module for calculating (h ij, 3 R, K ij, K i i ). 2 Module for calculating ( N, N i, γ ij, K ij, K, K ij, κ 0 ) and k i, K, κ. 3 Module for transformation to stereographic coordinates Integration 1 Module for the stereographic projections and source terms 2 Module for updating the evolution and algebraic equations Analysis 1 Module for implementing exact and perturbative test cases 2 Module for calculating errors and convergence rates

16 Numerics Numerical Integration: Runge-Kutta 2 nd and 4 th Order Flexible, can start and stop at any given radius Allows update of the algebraic equation inside the integrator Incorporates internally two types of CFL conditions given by: 1 Constant (linear) radial grid step h dqdp 2 Variable (logarithmic) radial grid step h rhodqdp Radial dissipation and stereographic interpolation ρ ρ + ɛð ð, and ρ ρ ɛ 4 ð 2 ð 2. Spatial Derivatives: Finite-Difference 2 nd and 4 th Order Radial derivatives: k k + O( hn 3 ), ρ ρ x k k Angular derivatives: ð module previously implemented.

17 Kerr-Schild Metric Testing The metric is given on the form: g ab = η ab + 2Hl a l b, H = M r 1 Exact Schwarzschild in centered cartesian coordinates (t, x i ), l a = ( 1, x i r, ), r = ρ = δ ij x i x j 2 Exact Schwarzschild in shifted cartesian coordinates (t, x = x x 0, ỹ = y y 0, z), l a = ( 1, x i r, ), r = δ ij x i x j 3 Boosted Schwarzschild in centered cartesian coordinates ( t = γ(t vx), x = γ(x vt), y, z), r = δ ij x i x j, l a = Γ b a l b, 4 Perturbed Schwarzschild in centered cartesian coordinates Ki i Ki i ( ) 1 + Y

18 1 Introduction Problems Approaches HyperSolID 2 Formulation Equations Variables Calculations Separation Transformation Projection 3 Implementation Algorithm Numerics Testing 4 Timeline Outline

19 Timeline Most of the code is already implemented The radial foliation, the separation of variables, the stereographic projection, the source terms, the 2 nd order integrator. The code passed two tests The centered and perturbed Schwarzschild tests prove a clean second order convergence, and long time stability. Work in progress Test the code with the shifted and the boosted Schwarzschild Implement and test the 4 th order radial integration Improve the angular derivative (Fourier or spherical harmonics) Implement and test 3 R, K ij, K i i, K ij, K