Conservative Evolution of Black Hole Perturbations with Time-Symmetric Numerical Methods Michael F. OBoyle1 Charalampos Markakis234 Lidia J. Gomes

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Conservative Evolution of Black Hole Perturbations

with Time-Symmetric Numerical Methods

Michael F. O’Boyle1, Charalampos Markakis2,3,4,*, Lidia J. Gomes

Da Silva2, Rodrigo Panosso Macedo5, Juan A. Valiente Kroon2

1Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illi-

nois 61801, USA,

2School of Mathematical Sciences, Queen Mary University of London, E1 4NS, Lon-

don, UK

3DAMTP, Centre for Mathematical Sciences, University of Cambridge, CB3 0WA,

Cambridge, UK

4NCSA, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

5Mathematical Sciences, University of Southampton, SO17 1BJ, Southampton, UK

Abstract. The scheduled launch of the LISA Mission in the next decade has called at-

tention to the gravitational self-force problem. Despite an extensive body of theoretical

work, long-time numerical computations of gravitational waves from extreme-mass-

ratio-inspirals remain challenging. This work proposes a class of numerical evolu-

tion schemes suitable to this problem based on Hermite integration. Their most im-

portant feature is time-reversal symmetry and unconditional stability, which enables

these methods to preserve symplectic structure, energy, momentum and other Noether

charges over long time periods. We apply Noether’s theorem to the master ﬁelds of

black hole perturbation theory on a hyperboloidal slice of Schwarzschild spacetime to

show that there exist constants of evolution that numerical simulations must preserve.

We demonstrate that time-symmetric integration schemes based on a 2-point Taylor

expansion (such as Hermite integration) numerically conserve these quantities, unlike

schemes based on a 1-point Taylor expansion (such as Runge-Kutta). This makes time-

symmetric schemes ideal for long-time EMRI simulations.

AMS subject classiﬁcations: 49S05, 49K20, 65M22, 65M25, 65M70, 70S10, 83-10, 83C25

Key words: time-symmetric integration, Hermite integration, black hole perturbation theory, hy-

perboloidal slicing

1 Introduction

The direct detections of gravitational radiation from compact binary coalescence by the

LIGO-Virgo-KAGRA (LVK) Scientiﬁc Collaboration in recent years has created a surge of

∗Corresponding author. Email address: c.markakis@qmul.ac.uk

http://www.global-sci.com/ Global Science Preprint

arXiv:2210.02550v1 [gr-qc] 5 Oct 2022

interest in gravitational wave (GW) science. Supplementing electromagnetic and particle

observations, we now have an additional window through which to view the universe

[17]. The events most likely to create observable GW events involve black holes, owing to

their compactness and strong curvature of spacetime [4,8,37]. However, the LVK detector

frequency band is only sensitive to events where the progenitors have comparable mass

[37], with the largest conﬁrmed mass ratio observed to date being ∼1:9 [2] †.

Another promising channel for observations are extreme-mass-ratio-inspirals (EM-

RIs) where a star or stellar-mass black hole orbits then plunges into a supermassive black

hole. Such events are expected to be regular occurrences in galactic centers and would

provide numerous astrophysical insights [4, 8]. A major complication in their study is

the difﬁculty entailed in accurately simulating the orbits of the smaller object and com-

puting the GWs emitted. Standard numerical relativity is poorly equipped to handle this

problem: the objects’ disparate masses creates two vastly different length scales, requir-

ing a ﬁne grid and small timesteps to accurately resolve, making long-time simulations

computationally intractable. A more promising route is the gravitational self-force pro-

gram, where the smaller object is modeled as a point mass that moves on a stationary

background spacetime. It sources linear perturbations that result in radiation reaction

and self-force effects [6,41]. It has been shown that, in a radiation gauge, these effects can

be derived by reconstructing the metric from curvature scalars [7, 29, 42, 52]. Thus, the

accurate evolution of scalar ﬁelds in curved spacetime has direct bearing on problems in

GW science and relativistic astrophysics.

Numerical relativity studies seem to favor explicit time-evolution schemes, like the

classical Runge-Kutta methods. Although they are easy to implement and well-studied,

they suffer from two drawbacks: they are conditionally stable, that is, CFL limited, and

known to violate energy conservation and symplectic structure in Hamiltonian systems.

In GW computations, it is vital to accurately track the energy a system loses to radiation,

and, with an explicit scheme, it is unclear a priori whether energy loss is due to radiative

loss or truncation error or other numerical dissipation. A preferable alternative is a so-

called geometric integrator which respects a qualitative feature of Hamiltonian dynamics,

like symplecticity or time-reversal symmetry [27, 50]. Time-symmetry is a particularly

appealing property, since Noether’s theorem relates time-translation symmetry to en-

ergy conservation. Moreover, such geometric methods often possess enhanced stability

properties. Geometric integrators have been considered in the context of developing a

numerical relativity based on the Regge calculus [23, 47], but the idea does not appear

to have been fully pursued. We argue that such schemes merit full consideration for the

reasons given above.

In previous work [35], we applied a class of time-symmetric methods derived from

Hermite integration to both the mechanics of a single particle and a classical wave equa-

tion sourced by a scalar charge. In the present work, we consider the master ﬁelds of

black hole perturbation theory (BHPT), showing that for each ﬁeld there are at least two

†A merger with an estimated mass ratio ∼1:26 was reported, but LVK concedes that this ratio is beyond the

capabilities of their models and reported the strong possibility of systematic errors [3].

conserved quantities derivable from Noether’s theorem (energy and U(1) charge) and

that Hermite methods numerically conserve both. We begin in Sec. 2 by presenting an

overview of method-of-lines numerics with Hermite integration, then proceed to the in-

tegration of classical ﬁelds in Sec. 3. We begin with the Schr¨

odinger ﬁeld of nonrelativistic

theory, which serves as a familiar example for outlining the machinery of more advanced

problems. We proceed to the massless Klein-Gordon ﬁeld governed by a scalar wave

equation in both ﬂat and Schwarzschild spacetimes. We ﬁnally examine gravitational

perturbations to the Schwarzschild spacetime in the Newman-Penrose formalism gov-

erned by the Bardeen-Press-Teukolsky (BPT) or Regge-Wheeler-Zerilli (RWZ) equations.

In each case, we examine which Noether-related constants are numerically conserved.

2 Time-Symmetric Evolution with Hermite Integration

We consider the problem of numerically approximating solutions to partial differential

equations (PDEs). Since the equations of BHPT are hyperbolic, we proceed using the

Method of Lines. That is, for a hyperbolic or parabolic PDE

∂tu(t,x) = ˆ

L(u(t,x)) (2.1)

where ˆ

Lis a (possibly nonlinear) spatial differential operator, we proceed by approxi-

mating the ﬁeld u(t,x)on a discrete spatial grid X={xi}N

i=0so that u(t,x)→u(t). The

components u(t,xi):=ui(t)of the vector u(t)are the values of the ﬁeld evaluated at the

gridpoints. Heuristically, this converts the problem from a PDE in space-time variables

(t,x)to a system of coupled ordinary differential equations (ODEs) in one time variable

dt =L(u). (2.2)

where the matrix operator Lcouples the set of ODEs. In this section, we will outline a

method for evolving such systems via numerical integration schemes symmetric under

time-reversal.

2.1 Hermite Integration

Using the fundamental theorem of calculus, the differential equations (2.2) can be con-

verted to a system of integral equations,

u(tn+1) = u(tn)+Ztn+1

f(t)dt, (2.3)

with the integrand f(t) = L(u(t)) treated as a function of time t. The problem has thus

been reduced to evaluating the time integral in Eq. (2.3).

2.1.1 1-point Taylor expansion

Integrating a (1-point) Taylor expansion of f(t)about the initial time tnyields the approx-

imant

Ztn+1

f(t)dt =

∑

m=1

∆tm

m!f(m−1)

n+Rl(2.4)

with remainder

Rl=∆tl+1

(l+1)!f(l)(t),t∈[tn,tn+1]. (2.5)

Here, we denote the m-th derivative of f(t)at t=tnby

f(m)

n=dmf(t)

dtmt=tn

. (2.6)

The time derivatives (2.6) may be determined exactly by recursively applying the chain

rule, f(m)=∂f(m−1)

∂u˙

u, with the last term substituted from the equation of motion (2.2). This

results in a single-step Taylor method. Alternatively, the derivatives (2.6) may be treated

as constant polynomial coefﬁcients and eliminated by evaluating the Taylor approximant

of f(t)at multiple points, resulting in a multi-step method, such as Runge-Kutta. These

two approaches are equivalent for linear systems. In any case, it is evident from Eq. (2.4)

that Runge-Kutta methods or 1-point Taylor expansions violate time-symmetry (that is,

Z2symmetry under time-reversal, tn↔tn+1,dt→−dt) and fail to preserve the symplectic

structure or Noether charges of Hamiltonian systems.

2.1.2 2-point Taylor expansion

A time-symmetric integration scheme can be obtained by approximating f(t)with a 2-

point Taylor expansion or, equivalently, a 2-point Hermite interpolant: an osculating poly-

nomial constructed to match the values of fand its derivatives at the endpoints tnand

tn+1. Integrating this osculating polynomial from tnto tn+1approximates the integral in

Eq. (2.3). This procedure is detailed in [35]. For the present work, we quote the end result.

Let us denote the m-th order time-derivative of f(t)at time tnby

f(m)

n=dmf(t)

dtmt=tn

. (2.7)

Integrating a Hermite interpolating polynomial which osculates derivatives up to order

l−1 yields the generalized Hermite rule [20, 30]:

Ztn+1

f(t)dt =

∑

m=1

clm ∆tmf(m−1)

n+ (−1)m−1f(m−1)

n+1+Rl(2.8)

with the expansion coefﬁcients given by

clm =l!(2l−m)!

m!(2l)!(l−m)!(2.9)

and the remainder given by

Rl= (−1)l(l!)2

(2l+1)!(2l)!∆t2l+1f(2l)(t),t∈[tn,tn+1]. (2.10)

Neglecting the remainder term, one can approximate the integral by summing lterms

on the right side of Eq. (2.8). The most important feature of this formula is its Z2sym-

metry under time-reversal (tn↔tn+1,dt →−dt). In addition, the remainder term scales

like ∆t2l+1: although the formula only contains terms up to ∆tl, it is accurate to O(∆t2l).

Moreover, the numerical pre-factor in Eq. (2.10) decreases much more rapidly with in-

creasing lcompared to a 1-point Taylor expansion (cf. [20, 30]). That is, even if we com-

pare methods of the same order, the truncation error in a 2-point Taylor expansion is

several orders of magnitude lower than that of methods based on a 1-point Taylor expan-

sion (such as the usual Runge-Kutta methods).

In this work, we will mainly demonstrate conservation properties of second- and

fourth-order time-symmetric integration rules, so we state them now. The choice l=1

yields the familiar trapezium rule,

Ztn+1

f(t)dt =∆t

2(fn+fn+1)+O(∆t3), (2.11)

which is accurate to second order. The choice l=2 yields the Hermite rule,

Ztn+1

f(t)dt =∆t

2(fn+fn+1)+ ∆t2

12 (˙

fn−˙

fn+1)+O(∆t5), (2.12)

which is accurate to fourth order. Here, the overdot indicates a time derivative. The

choice l=3 yields Lotkin’s rule [32]. Higher order generalizations can be obtained by

substituting l=4, 5, ... into Eq. (2.8) as detailed in [35].

If the schemes (2.8) are applied to the integral equation (2.3), an implicit scheme is ob-

tained to solve for u(tn+1). Moreover, since it is an implicit multi-derivative method of the

kind studied by Brown [12,13], it is unconditionally stable. That is, there is no Courant limit

on the timestep ∆t. And, as a time-symmetric method, it has been shown to numerically

conserve the energy and symplectic structure of Hamiltonian systems [35].

2.2 Application to Systems of Partial Differential Equations

2.2.1 Method of lines with time-symmetric discretization

Although they possess desirable theoretical properties, implicit methods are generally

require numerically solving nonlinear algebraic equations at every time step. If the origi-

nal PDE system is linear, then it is possible to construct an explicit evolution scheme from

these methods. We discuss how to do so now.

If the time integral (2.3) is approximated by the trapezium rule (2.11), we have

un+1=un+∆t

2[L(un)+L(un+1)]. (2.13)

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ConservativeEvolutionofBlackHolePerturbationswithTime-SymmetricNumericalMethodsMichaelF.O'Boyle1,CharalamposMarkakis2,3,4,*,LidiaJ.GomesDaSilva2,RodrigoPanossoMacedo5,JuanA.ValienteKroon21DepartmentofPhysics,UniversityofIllinoisatUrbana-Champaign,Urbana,Illi-nois61801,USA,2SchoolofMathematicalScienc...

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Conservative Evolution of Black Hole Perturbations with Time-Symmetric Numerical Methods Michael F. OBoyle1 Charalampos Markakis234 Lidia J. Gomes

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