Cauchy-characteristic extractions (CCE) avoids the errors due to extraction at finite worldtube. The Cauchy and the characteristic approaches have complementary strengths and weaknesses. Unification of the two methods is a promising way of combining the strengths of both formalisms

Babiuc-Hamilton, Maria

English

Marshall University

Marshall UniversityMarshall Digital ScholarPhysics Faculty Research PhysicsSummer 6-15-2009Cauchy-Characteristic Patching with ImprovedAccuracyMaria 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-Hamilton, M. C. (2009, June). Cauchy-characteristic patching with improved accuracy. Slides presented at the 12th EasternGravity Meeting, Center for Computational Relativity & Gravitation, Rochester Institute of Technology, Rochester, NY.Cauchy-Characteristic Patching with Improved Accuracy12th Eastern Gravity MeetingCenter for Computational Relativity & GravitationRochester Institute of TechnologyRochester, NY, June 15-16, 2009Maria C Babiuc-HamiltonMarshall University, WVIntroduction• Cauchy-characteristic extractions (CCE) avoids the errors due to extraction at finite worldtube• The Cauchy and the characteristic approaches have complementary strengths and weaknesses. • Unification of the two methods is a promising way of combining the strengths of both formalisms.Advantages• Avoids the errors due to gravitational waveform extraction at finite worldtube.• The grid domain is exactly the region of the waves propagation (no artificial boundary).• Gives the waveform and polarization state at infinity (no ongoing radiation). • Offers flexibility and control in prescribing initial data (very little gauge freedom).• Combine Cauchy & Characteristic methods.The Characteristic Method• Extract characteristic data at inner worldtube from Cauchy evolution,• Propagate the  characteristic solution along the outgoing light cones,• Extracts the waveform at infinity for each retarded time.The Stereographic Module• Newman-Penrose eth-formalism on the sphere• Numerical noise introduced by inter-patch interpolation.• Two improvements: Circular boundary,  4th order derivatives. qAqB DAU B = ∂ UHigher Order Approximations• 4th order approximations in finite differences.• 1st and 2nd derivatives of the 5th order, 3rdderivative of the 7th order Lagrange polynomial. D1F(xi) =F(xi−2) − 8F(xi−1) + 8F(xi+1) − F(xi+2)12∆x D2F(xi) = −F(xi−2) −16F(xi−1) + 30F(xi) −16F(xi+1) + F(xi+2)12∆x 2 D3F(xi) =F(xi−3) − 8F(xi−2) +13F(xi−1) −13F(xi+1) + 8F(xi+2) − F(xi+3)8∆x 3Errors in Angular Derivatives• 2D scalar wave propagation on the sphere:• Convergence rates for errors in −∂t2Φ + ∂ ∂ Φ = 0, Φ = cos ωt( )Ylm, ω = l l +1( ) ∂ 1Φ,∂ 2Φ and ∂ 3Φ.Error N=80 N=120 N=160 N=200 N=2403.99 4.04 4.11 4.35 4.853.99 4.07 4.24 4.80 3.953.94 3.98 3.95 3.92 3.86 ξ(∂ 1Φ) ξ(∂ 2Φ) ξ(∂ 3Φ)News for a Linearized Test• Bondi News function:• Linearized vacuum Bondi-Sachs solutions to Einstein equations on Minkowski background h = (l −1)l(l +1)(l + 2) 2Ylm Re(hl (r)eiνu) N = N+ + iN× = ∂th+ + ∂th×Alternate Method:  Ψ4• The Newman-Penrose Weyl component ψ4 Ψ4 = l∂N∂u+ O(l2)News and Comparison• Weyl tensor extraction Nψ is slightly more accurate than the News function extraction N. Ψ = N,u, NΨ = N u= 0 + Ψdu0u∫The Patching Scheme• Cauchy and characteristic evolution are patched in the vicinity of a worldtube,• Characteristic data is provided by Cauchy evolution at worldtube B,• Free initial characteristic data is given on the initial null hypersurface N,• All is embedded in the Cauchy evolution.Online Extraction• Sets the Cartesian coordinates on the sphere on which to extract the Cauchy data,• Interpolates the Cauchy metric, lapse, shift, and spatial derivatives of the metric on the sphere,• Calculates the Jacobeans from Cartesian to affine null metric, and from the affine to Bondi metric,• Calculates the boundary data and puts it on the worldtube with a Taylor expansion to be evolved,• Advances to the next time level and repeat.Towards a Versatile Extraction• CCE does not have to run simultaneously, if the data is given on the world tube, before extracting.• Steps towards a new IO interface for Extraction: Read from file Cauchy data in Cartesian coordinates between two determined radii, at a chosen resolution,  Convert data into a set of analytic functions using a Chebyshev and spherical harmonic decomposition,  Take the analytic functions and feed them to the extraction module (no interpolation necessary), Evolve the data and compute the waveform at infinity.Chebyshev Spherical Harmonic Decomposition• Normalization for Chebyshev polynomials:• Change the integration limits, add Ylm and do the coordinate transformation (r,θ,φ)->(x,y,z):• Recover the analytic function: wUm−1+1∫ Undt = δmn π2 ⇒ fn =2πFw−1+1∫ Undt, w = 1− t 2 flmn =2πdtdrwr2R1R2∫ Ylm* UnFdxdydz F = flmnYlmUnn∑m∑l∑Implementation• Define a mask between two relevant radii (R1, R2)• Read the functions in (x,y,z) coordinates,• Compute the expansion coefficients flmn as a sum over the masked points,• Reconstruct the function,• Populate the worldtube. flmn = mask Flmni, j ,ki, j,k∑ ∆x∆y∆zFurther Improvements• Include higher order approximations in the post-processing of the news function• Improve the characteristic boundary by changing the data on the inner worldtube from Dirichlet to Sommerfeld• Produce a Cactus CCE module for wave extraction that will be freely available to the numerical relativity community

Cauchy-Characteristic Patching with Improved Accuracy

https://mds.marshall.edu/cgi/viewcontent.cgi?article=1071&context=physics_faculty

Cauchy-Characteristic Patching with Improved Accuracy

Abstract

Similar works

Full text

Available Versions

Marshall University