Discontinuous Galerkin Method for Computing Perturbations of a

Schwarzschild Spacetime

Scott FieldDepartment of Physics

Brown Universitywith Jan Hesthaven (adviser) and Stephen Lau

Organization of PresentationOrganization of Presentation

• Overview of Perturbation Treatment of the Extreme Mass Ratio Binary (EMRB) Problem

• Discontinuous Galerkin Method • Numerical Scheme for Perturbation

Equations• Results• Conclusion and Future Work

Extreme Mass Ratio Binary

Physical RelevancePhysical Relevance

• EMRBs expected to be primary source of gravitational waves detected by LISA– “Small” star of mass orbiting a super-

massive black hole of where • Motivation for considering new high-order

(in this particular case spectral) numerical methods: Construction of quality gravitational waveforms in the shortest amount of time.

pm1/ < <= MmpµM

Background and AssumptionsBackground and Assumptions

• Point particle in stable orbit of SMBH– Schwarzschild– Particle follows a time-like geodesic in equatorial plane– Stress energy tensor

• Particle causes small metric perturbations– Metric– Perturbations can be combined into master-functions

which evolve according to

)]([4 τδτ βαα β zxuug

dmTSchp −

−= ∫

222122 Ω++−= − drdrffdtds rMf /21−=

α βh⇒+= α βα βα β hgg Sch

α βα β πδ ThGLin 8)( =

)(),()(),(),())(( ////22 rprrtFrprrtGtxrV rPAPAPAPA

xt −∂+−=−∂+− ∂ δδψ

Equations and QuantitiesEquations and Quantities

• In the distant wave-zone and at BH horizon:

• Metric perturbations can be reconstructed everywhere in space

2||)!2()!2(

641

lmlm llE ψ

π

−+= lmlmlm l

limL ψψπ

*

)!2()!2(

64 −+=

lmAlm

Plm

mlx Yi

ll

rihh 2

,2

)!2()!2(

21

−≥

+ +−+=− ∑ ψψ

Discontinuous Galerkin Method

Numerical Simulations of EMRBsNumerical Simulations of EMRBs• Numerous proposed methods, such as…

– Frequency-domain• Fourier decomposition• Becomes numerically expensive when one must sum

over many modes – e.g. eccentric orbits– Time-domain

• Typically spatial derivatives are approximated by finite differences

– Errors fall off as a power– Long time integrations can produce phase errors in low order

methods• Distributional source term can be troublesome• Large computational domain-size

– Waveform/BC issues

DG: A Hybrid of MethodsDG: A Hybrid of Methods

• Finite element method: if Dk = Ω and basis functions are “coupled” (ie span multiple elements) we get a finite element method

•F

• Legendre collocation, “Pseudospectral,” on each element

• When basis functions are constants, dG formally becomes a finite volume method

Figure created by Oleg Alexandrov

Spectral and Finite Difference ErrorsSpectral and Finite Difference Errors• 1D advection

equation with speed 2pi, on a domain 0 to 2pi. Integration with RK4

)2sin()( txexu π−=

C

Spectral Methods for Time Dependent Problems by Jan Hesthaven & David Gottlieb

Advection: Phase ErrorsAdvection: Phase Errors

• When evolving hyperbolic PDEs for long times, phase errors can become significant

• An example with the advection equation

Numerical solution propagate at wrong speed

)exp()0,( :

0

ikxxICx

ct

=

=∂

∂+∂

∂

ψ

ψψ

)](exp[),( ctxiktx −=→ ψ

)exp( :

0

jj

jNumj

ikxIC

cDt

=

=+∂

∂

ψ

ψψ

)](exp[)( tcxikt numjj −=→ ψ

Advection: Phase ErrorsAdvection: Phase Errors

• If using second order central difference

we have a phase error of

– Typically phase errors scale as – To achieve small phase error one must:

• Run for short times• Use a fine grid• High order method

)(211

xD jj

jNum ∆−

= −+ ψψψ

6)(sin1)exp()exp(

2xkktcxk

xkktctikcikct num∆=

∆∆−=−−−

orderxkt )( ∆∝

Advection: Phase ErrorsAdvection: Phase Errors

• Using a 4th order Runge-Kutta time integrator, evolve a Gaussian packet using first and second order approximations to

• In each example we have chosen x∂

∂

5.~x∆

11stst Order Upwind Differencing Order Upwind Differencing

22ndnd Order Central Differencing Order Central Differencing

Discontinuous GalerkinDiscontinuous Galerkin

• Domain Ω approx. by local elements Dk

• Local solution is a linear combination of basis functions

– Form a basis for space of polynomials in Dk of degree at most N

• Global solution is a direct sum of local solutions

nP

)( ),()()(ˆ),( :00

xltxuxPtutxuDx ki

N

ii

khn

N

n

kn

kh

k ∑∑==

==∈

),(),(),(1

txutxutxu kh

K

kh =⊕=≈

K

k

kh D

1=

=Ω≈Ω

)(0: kNn DPNnPSpan =≤≤

Discontinuous GalerkinDiscontinuous Galerkin• Given an operator L such that can be

cast in integral form on elements Dk

– Require of numerical solution uh

– Galerkin condition: test and basis functions taken to be the same

To resolve: coupling between elements?

∫ ≤≤=kD

kn

kh NndxvLu 0 ;0)(

0=Lu0)( =∫ vdxLu

kD

N

Nodal Discontinuous Galerkin Methods by Jan Hesthaven and Tim Warburton

Numerical FluxNumerical Flux

• To couple elements first perform IBPs

• Numerical flux – and are the exterior and interior solutions– passes information between elements, ensures

stability of scheme, implements boundary conditions– Choice of related to dynamics of PDE system

• Consistency condition:

• Example:– Central flux

vdxufndxvufuvvdxufu hDD

xkk

tD

kx

kt

kkhh

khh

))((ˆ))((0))(( *⋅−=∂−∂→=∂+∂ ∫∫∫∂

),()( ** −+=hh

uufuf h+h

u −h

u

)(*huf

),()( *hhh uufuf =

2)()()(*

+− += hhh

ufufuf

Example: Advection with a Distributional Source Term

Problem StatementProblem Statement

• Each sector of the perturbation equations similar in spirit to 2 copies of advection

• Equation:

• Domain:• BC: • Initial data:• Jump condition:• Sends information G(t) off to the right at speed c

)/()(),( 00 cxtGxtx −Θ=ψ

0)()(1 =−∂

∂+∂

∂ xtGxtc

δψψ

)()0()0( tG=− −+ ψψ

baba <<=Ω 0 s.t. ],[0),( =taψ

) , ,( paritym

Numerical SchemeNumerical Scheme

• Basis space: Lagrange interpolating polynomials of order N on defined on each element

– Legendre-Gauss-Lobatto nodal points η • Integrate with fourth order Runge-Kutta• Numerical flux:

– PDE shifts solution to the right, so use an “upwinded” form

∏≠= −

−=

N

ijj ji

ji

xxl

0

)(ηηη

)2ˆ

2ˆ

()(*−−−+++ ⋅++⋅+= ψψψψψ nncf h

A Generalized DG (GDG) MethodA Generalized DG (GDG) Method• GDG extends DG to solutions (analytically)

discontinuous at an interface• Idea: treat the δ function in the PDE as an

additional numerical flux term in an appropriate way, ie mimic the physics of the PDE– Require usual delta property over Ω

– Freedom to choose how to “split” the delta function between adjacent elements

K. Fan, W. Cai, X. Ji, A generalized discontinuous Galerkin (GDG) method for Schrodinger equations with nonsmooth solutions, J. Comp. Phys., 227 (2008) 2387-2410.

)0()()( vdrxvx =∫Ω

δ

)0()0()0()()()()()()(11

vbvavdrxvxdrxvxdrxvxiiii DDDD

=+=+= ∫∫∫++

δδδ

Treating an Advected Treating an Advected δδ Source Source

• Upwind the source:

• GDG principle suggests we add a flux term to the existing flux term on the right hand side of the interface

δ*,,1)( LicG +

0*)]()()[( =∫ dxxtcGxliD

δ

)0()()]()()[(1

ltcGdrxtcGxliD

=∫

+

δ

*)],([ txcψ

Comparison With Analytic SolutionComparison With Analytic Solution

• Analytic solution is– Choose c = 2, G(t) = cos(t), final time = 15

)()(),( cxtGxtx −Θ=ψ

Exponential convergenceExponential convergence • Final time = 30• 4 domains on [-5,5]• Time-step set below CFL

requirement of largest N

44thth order RK order RK • Final time = 30• 30 Domains on [-5,25]• Polynomial Order = 10

A Numerical Schemefor the Perturbation Equations

Issues and ResolutionsIssues and Resolutions

• Particle must be located at an interface for spectral accuracy to be maintained– Coordinate transformation

• The derivative of a delta function term differentiates the test functions– Rewrite equations in full first order

• Non-trivial boundary conditions– Exact outgoing boundary conditions

• Computing waveform at null infinity– Use flat space extraction (Schwarzschild

solution asymptotically flat)

Rewriting the Equations in First Rewriting the Equations in First Order SystemOrder System

• As first order system, and specializing to

• Using properties of distributions, one can show this is the first order form of the original equation

)()(

)()()(

pxt

pxt

t

xxtJxxtJrV

−+Π− ∂=Φ∂

−+−Φ− ∂=Π∂Π−=∂

Π

Φ

δδψ

ψ

(t)J |][ (t)J |][

)(),()(),())(( 22

Φ=Π= =∂=− ∂

−∂+−=−∂+− ∂

pp xxxxxt

prpxt rrrtFrrrtGrVψψ

δδψ

pp rtr =)(

Semidiscrete Form Semidiscrete Form

upwind numerical flux (generalization of advection example)

−−+

−

Π−+

ΠΦ− ∂=

ΦΠ∂

Π

Φ

)(),()(),(

0

0

0

rprtrJrprtrJVxt

δδψ

ψ

)(),(),(0

xltxutxu ki

N

ii

kh

kh ∑

=

=

dxuHxldxJxldxufxlndxxlufdxtxuxlkkkkk D

hiD

iD

hiD

ixhD

hti ∫∫∫∫∫ ++⋅−∂=∂∂

)()()()()(ˆ)()(),()( *

JuHufu xt

++− ∂=∂ )()(

)(*huf

LGL Nodal Points +

Treating the Source TermsTreating the Source Terms

• Diagonalize the sourced sector of the system

• GDG suggests adding to on the RHS of the interface and to on the LHS of the interface

• Numerical flux at particle interface is changed as

λωλω

−=Π+=Φ

2/)()(

/2)()(

pxt

pxt

xxJJxxJJ

−−=∂−∂

−+=∂+∂

ΦΠ

ΦΠ

δλλδωω

δδ ω *,*, )())(5(. RRJJ =+ ΦΠ*)(ω

δδ λ *,*, )())(5(. LLJJ =− ΦΠ*)(λ

δδ λω *,*,** )()( LR ++Φ→Φ δδ λω *,*,** )()( LR −+Π→Π

Boundary ConditionsBoundary Conditions

• Non-reflecting BC– Reduces domain size, especially useful for eccentric

orbits where many periods required– Sommerfeld BC works well near BH horizon as– At the right boundary Sommerfeld fails as

instead

– depends on form of potential – Approximated using known techniques to satisfy a

specifiable error tolerance, here

Stephen Lau. Rapid Evaluation of Radiation Boundary Kernels for Time-domain Wave Propagation on Blackholes. gr-qc/0401001

MxeV 2/~2/1~ rV

),,()( ψψ bxt rtF=∂+∂

1010−<ε

),,( ψbrtF

Sketch of BC in Flat 3+1 SpaceSketch of BC in Flat 3+1 Space

• For given ell, write down an arbitrary outgoing solution in the Laplace frequency domain

• Apply Sommerfeld operator• Rewrite result in form

• Taking the inverse Laplace transform

• Theorem: In the time-domain, the kernel is a sum of exponentials

rs ∂+

),(),(ˆ1),()( rsrsr

rss outoutr ΨΩ=Ψ∂+

'),'(),'()(0

dTrTrTTC b

T

brt ∫ Ψ−Ω=Ψ∂+∂

( )∑=

≅Ωd

kkk t

1,, exp βγ

Boundary ConditionsBoundary Conditions

Long-time bound on error due to boundary conditions. Final-Time = 100M

Waveform MatchingWaveform Matching

• Want to “read off” at null infinity • Recording at introduces an error • suggests that• At right boundary we match waveforms as ( )

• This is an ODE sourced by and approximates the waveform at infinity – Reduces the error to

)( 3SpaceFlat A/P −+= rOVV SpaceFlat A/P ~ψψ

)( 2−xO

)(3)(3)(~ 2)1()2(SpaceFlat

2A/P

2 bb

bb

b xtfx

xtfx

xtf −+−+−=== ψψ

A/PψA/Pψ bx )( 1−

bxO

2=

)()2( xtf −A/P2=ψ

Initial DataInitial Data

• Typically trivial initial data supplied, but this is inconsistent with the PDEs

• “Switch-on” the source terms by multiplying them by a function which smoothly interpolates from 0 to 1– We used [ ]1))((

21 +− τδ terf

Initial Data: 1+1 Wave Equation Initial Data: 1+1 Wave Equation

• Solution to 1+1 wave equation with distribution source term– Have analytic

solution• Without smoothing 4th

order temporal convergence is abruptly lost (red). It is recovered with a smoother (blue)

Features of ImplementationFeatures of Implementation

• Spectral global accuracy, even at particle– Potentially useful for incorporating a self-force– Excellent phase resolution for long integrations

• Reduced computational domain without sacrificing accuracy of waveform or introducing reflection– Very useful for problems without periodicity

• Particle trajectory need not be a priori• No length scale associated with particle

Results

Results: Circular OrbitResults: Circular Orbit0)0,( ,]1000 ,200[* ,9456.7 ,1 ,)2,2(),( =−=Ω=== xurMm pBH

Energy and Angular Momentum Energy and Angular Momentum LuminosityLuminosity

lmmlm llimL ψψ

π

)!2()!2(

64 −+=

2||)!2()!2(

641

lmlm llE ψ

π

−+=

Results: Eccentric OrbitResults: Eccentric Orbit

2436.4887153.22

0682.52873.8

63517.

max

min

==

==

=−=

ϕTTrr

latussemityeccentrici

r

0)0,( ,]1000 ,200[* ,1 ,)2,2(),( =−=Ω== xuMm BH

Eccentric Orbit WaveformsEccentric Orbit Waveforms )2,2(),( =m

Eccentric Orbit QuantitiesEccentric Orbit Quantities

Selected energy and angular momentum flux calculated at null infinity

• Averages computed according to

r

T

Tfm

fm TTTdtE

TTE

f

4 10

0 0

=−−

> =< ∫

What's Next?What's Next?• Short term:

– Calculate self force• Done for and when orbit quasi-circular• For quasi-circular inspiral orbits would like to calculate

waveforms due to dissipation• Long term:

– Exact BC for left side of computational domain • Arbitrarily short domain size

– Could calculate up-to ell=25 for 200 orbits of particle initially at r=7.9M in ~3 weeks

» On Dell laptop, coded in Matlab» Optimistically on a GPU with python code one might

achieve a speed up factor of 50 or more

– Waveform extraction• Weak link, suggestions?

lmE lmL

Summary and KerrSummary and Kerr• Purposed a high order DG method for EMRBs

– Equations as a first order system, exact outgoing radiative BCs, waveform matching, and smooth startup

– Results agree with literature to stated accuracy– Global spectral accuracy

• EMRB problem for Kerr solution– 2+1 dimensional– 2D problems with distributional sources have been

considered in the problem by Fan et al