Results in developing two new methods to improve the accuracy of waveform extraction using characteristic evolution. Numerical method: circular boundaries, with angular dissipation in the characteristic code. Geometric method: computation of Weyl tensor component Y4 at null infinity, in a conformally compactified treatment. Comparison and calibration in tests problems based upon linearized waves

Babiuc-Hamilton, Maria

English

Marshall University

Marshall UniversityMarshall Digital ScholarPhysics Faculty Research PhysicsSpring 5-19-2009Towards Improved Accuracy of GravitationalWaves ExtractionMaria Babiuc-HamiltonMarshall University, babiuc@marshall.eduFollow this and additional works at: http://mds.marshall.edu/physics_facultyPart of the Physics CommonsThis Presentation is brought to you for free and open access by the Physics at Marshall Digital Scholar. It has been accepted for inclusion in PhysicsFaculty Research by an authorized administrator of Marshall Digital Scholar. For more information, please contact zhangj@marshall.edu,martj@marshall.edu.Recommended CitationBabiuc, M. C. (2008, August) Towards Improved Accuracy of Gravitational Wave Extraction. Slides presented at 2008 NumericalRelativity and Data Analysis Meeting, Syracuse, NY.Towards Improved Accuracy of the Gravitational Waves ExtractionMaria C. BabiucDepartment of Physics, Marshall UniversityHuntington, WV2008 Numerical Relativity and Data Analysis MeetingSyracuse, NY, Aug.11 2008Abstract• Results in developing two new methods to improve the accuracy of waveform extraction using characteristic evolution. • Numerical method: circular boundaries, with angular dissipation in the characteristic code.• Geometric method: computation of Weyl tensor component Ψ4 at null infinity, in a conformally compactified treatment.• Comparison and calibration in tests problems based upon linearized waves.Introduction• The artificial finite outer boundary present in Cauchy codes introduce two sources of error:• The outer boundary condition, • Waveform extraction at an inner worldtube.• The problem of proper boundary condition for a radiating system can be solved only by extension to Ι4 (conformal compactification).• Cauchy Characteristic Extraction (CCE) offers a means to avoid these errors.Introduction• The CCE code extends the solution to Ι+ by matching the interior Cauchy evolution to an exterior characteristic evolution.• The code uses the data on a worldtube provided by binary black hole spacetimes obtained with any Cauchy evolution codes, and computes the gravitational radiation reaching infinity in terms of the supplied boundary data.Sources of Error• Perturbative regime tests compares favorably CCE with Zerilli extraction, and show CCE advantage at small radii.• Nonlinear tests show CCE stable, but plagued by numerical error in the numerical postprocesing at null infinity.• Two ways: numeric and geometric, to improve the accuracy of the waveform.Ways to improve accuracy• Geometrical: computation of the asymptotic  of part of Ψ4 and comparison with the news N.• Numerical: improvement of intergrid interpolations between the patches smoothly covering the sphere. Comparison between:• The circular stereographic patching, • The cubed-sphere patching.• Alternatives: higher order finite difference approximations, adaptive mesh refinement.Characteristic Formulation• Based on a family of outgoing null hypersurfaces, from the worldtube to infinity, in Bondi-Sachs metric:• The Einstein equations Gµν=0 decompose into hypersurface, evolution and conservation equations. The evolution equation takes the form:• The code implements this as a second order finite difference scheme, all angular derivatives first order. ds2 = − e2β Vr− r2hABUAU B      du2 − 2e2β dudr − 2r2hABUB dudxA + r2hAB dxA dxB 2 rJ( ), ur − r−1V rJ( ), r( ), r = −r −1 r 2∂ U( )r + 2r −1eβ∂ 2eβ − r −1V( ), r J + NJ J = 12hABqAqB , qAB =12qAq B + q AqB( )Angular dissipation• Numerical dissipation is necessary to:– stabilize the intergrid interpolation error,– suppress the circular boundary high frequency error• The evolution equation takes the form:• We introduce angular dissipation in the retarded time uand radial r evolutions:• We dissipate also the hypersurface equations. ∂u 1− x( )Φ,x + Φ( )= S, x = r R + r( ), Φ = xJ, ∂u 1− x( )Φ,x + Φ( )→ ∂u 1− x( )Φ,x + Φ( )+ εuh3∂ 2W∂ 2∂u 1− x( )Φ,x + Φ( ) ∂u 1− x( )Φ,x + Φ( )→ ∂u 1− x( )Φ,x + Φ( )+ εxh3∂ 2W∂ 2Φ,uWaveforms at null infinity• Conformal Penrose compactification of Bondi metric:• Future null infinity I+ is at l=0. The Bondi mass (total energy), news N and Ψ40 (radiation power), are constructing from expansion of metric in powers of l.• H, HAB, cAB and LA are expansion coefficients.• One can require the Bondi coordinate to be inertial (Minkowsky) at I+ but it is not assumed: the waveform characteristic extraction is done in null coordinates.  ˆ gµν dxµdxν = − e2βVl3 − hABUAU B( )du2 + 2e2β dudl − 2hABU B dudxA + hAB dxA dxB l =1 r, ˆ gµν = l2gµν 2HC (A DB )LC + ∂uHAB − HAB DC LC = O(l)Calculation of the News• In an inertial conformal Bondi frame the News are :• where:• An explicit calculation leads to:• In inertial Bondi coordinates:• The general form is used, which is challenging because of second order angular derivatives of ω. ˜ gµν = Ω2gµν = ω2 ˆ gµν , Ω = ωl, QAB := ˜ gab I + = ω2HAB N = limΩ→012ΩQαQβ ˜ ∇ α ˜ ∇ βΩ N = 14e−2iδω−2e−2 H FαF β ∂u + L L( )cAB −12cAB DC LC + 2ωDA ω−1DB ωe2 H( )[ ]       H AB = F A F B + F A F B( )/2 F A = qA K +12 − q A J 12 K +1( ), Qβ = e− iδω−1F β + λ ˜ nβ N = 14QAQB∂ucABCalculation of Weyl tensor• Weyl tensor vanishes at I+ (asymptotic flatness)• The inertial radiation field in terms of code variables:• involves lengthy algebra. In inertial Bondi coordinates• However, general form is used, which is challenging because of third order angular derivatives of ω. ˆ nµ = ˆ ∇ µl, ˆ l µ∂µ = ∂l ˆ Ψ := − 12liml →01lˆ nµ ˆ m ν ˆ nρ ˆ m σ ˆ C µνρσ = −12Ψ 40, Ψ =12ω−3e−2iδ ˆ nµF A F B ∂µ ˆ Σ AB −∂A ˆ Σ µB − ˆ Γ µBα ˆ Σ Aα + ˆ Γ ABα ˆ Σ µα( )I + Ψ =14QAQB∂µ2cAB = ∂µ2∂lJ I + = ∂uN .Linearized Expressions• The general nonlinear representation of Ψ in terms of the computational variables reduces to a simpler form in first order perturbations off Minkowski background.• This provide a starting point to compare the advantages between computing the radiation via the Weyl component or the news function.  ω propagates across patches Ψ =12∂u2∂lJ −12∂uJ −12∂ L − 18∂ 2 ∂ L + ∂ L( )+ ∂u∂ 2H N = 12∂u∂lJ +12∂ 2 ω + 2H( ) 2 ˆ nα∂α logω = −e−2 H DA LA .Patching the Sphere• The nonsingular description of smooth tensor fields on the sphere requires more than one coordinate patch.• We consider two treatments: the stereographic, using 2 patches, and the cubed-sphere, using 6 patches.• In the stereographic approach, every point on the sphere is covered by at least one of the patches, and around equator, points are covered by two patches.• We implement the circular stereographic method, based on the composite-mesh method, where the overlap is reduced to a circular region around equator.Circular patches• Complex stereographic coordinates cover the sphere• Unit sphere metric in each patch:• All boundary points of one patch are interior points of another patch. The overlapping of the patches is key to the stability of method. The discretization is:• The active finite difference grid:• Stability requires that the interpolation stencil for one patch ghosts points lies below equator in other patch. ξN = qN + ipN = tan(θ /2)eiϕ , ξS =1/ξN qAB dxA dxB= 4P 2(dq2 + dp2), P =1+ q2 + p2, qA = P2(1,i), q2 + p2 =1 qi = −1+ (i − O −1)∆, p j = −1+ ( j − O −1)∆,1≤ i, j ≤ M +1+ 2O qi2 + p j2 ≤1+ (O − RE )∆ FS (ξS =1/ξN ) = FN (ξN )(−1)se−2isϕThe cubed sphere• Sphere covered by 6 coordinate patches, obtained by by projecting 6 faces of a circumscribed cube.• Recently applied to characteristic evolution gr-qc/0610019• For M2 stereographic grid points, there are πM2/4 grid cells inside equator on each hemisphere.• In the cubed sphere grid, with N2 points per patch, the entire sphere is covered by 6xN2 points. This gives:• The tests are run with M=100,120 for the circular patch, which correspond to N=51,61 for cubed-sphere, t=120.• We monitor the convergence and smoothness of error: ε(Φ) = Φnumeric − Φanalytic ∞ N 2 ≈ π /12( )M 2Comparison between circular and cubed methods• A test of 2D wave propagation on the sphere:• Allows direct comparison between the circular patches and the cubed-sphere methods, without characteristic evolution and Ψ4 and N computation.• Angular dissipation, necessary for the circular case:• Emphasis on the accuracy of the angular derivatives required by Ψ4 and N in the waveform extraction.  −∂t2Φ + ∂ ∂ Φ = 0, Φ = cos ωt( )Ylm, ω = l l +1( ) ∂t2Φ → ∂t2Φ + ε∆3D4∂tΦ, D4Φ = P 2 /4 D+qD−q + D+ pD+ p( )( )2ΦError in Φ and δ2ΦAlgrthm T=1.2 T=12 T=102 T=120circular 2.00 1.99 1.99 2.00cubed 1.99 1.97 1.98 1.99Algrthm T=1.2 T=12 T=102 T=120circular 2.02 1.95 1.99 2.01cubed 1.95 2.02 2.00 1.97Error in δ3ΦAlgrthm T=1.2 T=12 T=102 T=120circular 2.28 2.03 1.99 2.01cubed 1.11 0.88 2.01 1.96Our choice• For ε(Φ),  clear 2nd order convergence for both methods is observed. The  cubed sphere error is smaller than the stereographic error (1/3).• For ε(δ2Φ), the cubed sphere error is 2/3 the stereographic error. Again, 2nd order convergence.• For ε(δ3Φ) the cubed sphere method shows poor convergence at early times. • Until t=60, the cubed-sphere method has the largest error, but at the end, is 4/5 the stereographic error.• These results justify our choice of the circular patches stereographic method in the comparison of the news N and Weyl tensor Ψ4 extraction.Comparisons of News and Weyl tensor extraction• We base the test on a class of solutions in Bondi-Sachs form to the linearized vacuum Einstein equation on a Minkowski background:• Solution: well-behaved at I+ and well-defined at r>r0>0 J = (l −1)l(l +1)(l + 2) 2Ylm Re(Jl (r)eiνu) N = Re eiνu limr→∞−l(l +1)4Jl −iν2r2Jr,l      + eiνuβl      (l −1)l(l +1)(l + 2) 2Ylm Ψ = N,u, NΨ = N u= 0 + Ψdu0u∫ J2(r) =24β0 + 3iνC1 − iν3C236+C14r−C212r3, J3(r) =60β0 + 3iνC1 + ν4C2180+C110r−iνC26r3−C24r4 C1 = 3 ⋅10−6, C2 =10−6, β0 = i ⋅10−6Test results for J• Runs with circular patch, circular without dissipation, and the original square patch methods. The plots show that error increases with x and is maximum at I+. Also, that angular dissipation reduces the error.Convergence for JVrbl circle crnods squareT=1 2.01 2.01 2.01T=10 1.95 2.00 1.99T=90 2.07 1.96 2.00T=100 1.92 2.01 1.99Vrbl circle crnods squareT=1 2.02 2.02 2.02T=10 1.99 1.99 2.00T=90 2.02 2.02 2.04T=100 2.00 2.00 1.99Surface Plots for JTest results for the news:N (left) and NΨ (right)Vrbl circle crnods squareT=1 2.04 2.04 2.04T=10 2.04 1.99 2.04T=90 2.01 2.01 2.06T=100 1.98 2.00 1.93Vrbl circle crnods squareT=1 2.08 2.08 2.08T=10 2.09 2.05 2.10T=90 2.05 2.00 2.06T=100 1.98 2.01 1.93Surface Plots for N• Effectiveness in applying dissipation. Slightly more jaggedness near the equator for the circular patches is overbalanced by the relative smallness of its error.Surface Plots for NΨ• The error in NΨ is slightly smaller, otherwise there is little difference between N and NΨ.Vrbl circle crnods squareN 2.25x10-9 3.32x10-9 2.90x10-9NΨ 1.71x10-9 2.75x10-9 2.32x10-9Conclusions• For linearized case no method is clear winner.• The news calculated on a circular patch had lower error than that on a square patch (30%). • Weyl tensor extraction is slightly more accurate than news function extraction (24%).• Very small fractional error (0.1%) in metric J.• The corresponding averaged error in the NΨand N was 4% for the circular patch runs and the maximum error at the equator was 9%.Conclusions• All errors were second order convergent. • The  errors did not vary appreciably (30%) with the choice of discretization method.• Intrinsic difficulty in extracting waveforms due to the delicate cancellation of leading order terms in the metric and connections.• The excellent accuracy that we find for the metric suggests that perturbative waveform extraction must suffer the same difficulty.Conclusions• Waveforms are not easy to extract accurately.• The convergence of the error is a positive sign that higher order finite difference approximations might supply the accuracy needed for realistic astrophysical applications. • Whether the advantages the new methods proposed here prove to be significant will depend upon the results of future application in the nonlinear regime.Thanks• Collaboration: Nigel Bishop, Bela Szilagyi, Jeff Winicour• Strategies for the Characteristic Extraction of Gravitational Waveforms (arXiv:0808.0861)• M. C. Babiuc, N. T. Bishop, B. Szilagyi, J. Winicour, submitted to Physical Review D• Open Source CCM: we work toward making the characteristic extraction module available to the numerical relativity community.

Towards Improved Accuracy of Gravitational Waves Extraction

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Towards Improved Accuracy of Gravitational Waves Extraction

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