A spectral collocation approximation of one-dimensional head-on collisions of black-holes Jae-Hun Jung ∗ Department of Mathematics, University of Massachusetts at Dartmouth North Dartmouth, MA 02747-2300, USA
Gaurav Khanna, and Ian Nagle Department of Physics, University of Massachusetts at Dartmouth North Dartmouth, MA 02747-2300, USA
Abstract The inhomogeneous Zerilli equation is solved in time-domain numerically with the Chebyshev spectral collocation method to investigate a radial infall of the point particle towards a Schwarzschild black hole. Singular source terms due to the point particle appear in the equation in the form of the Dirac δ-function and its derivative. The main objective of the current work is to show that the spectral collocation method yields an accurate approximation of head-on collisions of black holes even with singular source terms. For the approximation of singular source terms, we use the direct derivative projection method proposed in [9] without any regularization. The waveforms are evaluated with time and the gravitational ringdown is measured. We also compare the results of the spectral collocation method with those of the explicit second-order central-difference method. The numerical results show that the spectral collocation approximation with the direct projection method is accurate and stable for long time integration. Key words: Head-on collisions of black holes, Singular sources, Direct projection method, Dirac δ-distribution, Chebyshev spectral collocation method, Finite difference method.
∗ Corresponding author. Tel:+5089998342; fax: +5089106917 Email addresses:
[email protected] (Jae-Hun Jung),
[email protected] (Gaurav Khanna).
Preprint submitted to Elsevier Science
22 May 2008
1
Introduction
The capture of a stellar mass compact object (a black hole or star) by a supermassive black hole occurs commonly in the nuclei of most galaxies. As the compact object spirals into the black hole, it emits gravitational radiation which would be detectable by space borne detectors such as NASA/ESA’s LISA [1]. This paper is about the calculation of emitted gravitational radiation from such events, often referred to as extreme-mass-ratio binary inspirals or EMRIs. Because of the extreme mass-ratio, the small companion of a supermassive black hole can be modeled as a point particle, and the problem can be addressed using black hole perturbation theory. Moreover, as a first approximation, we will let the point particle follow geodesics in the space-time of the central black hole. This type of approach has been taken by several researchers in the past using a frequency-domain decomposition of the master equation (for a recent review see [5]). That approach has had many remarkable achievements, specifically the accurate (to 10−4 ) determination of the energy flux of gravitational waves for many classes of geodesic orbits. However, for orbits of high-eccentricity and parabolic orbits, frequency-domain computations typically result in long computation times and large errors. It is here that a time-domain based approach would be invaluable. Finite-differencing based, time-domain methods for such evolutions were only recently developed and they suffer with a relatively poor accuracy of about 1% [2,21]. The main source of error in these methods is the approximate modeling of a point particle, i.e. a Dirac δ-function, on a finite-difference numerical grid. Various approaches to this issue have been attempted, including “regularizing” the Dirac δ-function using a narrow Gaussian distribution [2] and also using more advanced discrete-δ models [4,21] with only limited success. An alternate approach to the time-domain based solution of the EMRI problem is to use the spectral method instead of the finite-differencing approach discussed above. It is well known that the spectral method yields so-called spectral convergence for the approximation of smooth problems. That is, the accuracy of the spectral method is by an exponential order rather than an algebraic order which is the typical order for the finite difference method. The spectral method, however, loses its spectral accuracy when discontinuous or singular problems are considered. Convergence of the spectral approximation for discontinuous problems is only O(1) in its L∞ norm. For the approximation of singular source terms, the spectral Galerkin approach yields an efficient way because the Galerkin projection of the Dirac δ-function to any polynomial spaces can be easily and exactly evaluated. The spectral Galerkin approximation is, however, only limited to O(1) convergence near the singularity or the local jump discontinuity. The spectral collocation method also suffers from 2
this O(1) convergence and this is well known as the Gibbs phenomenon. The Gibbs oscillations found in spectral approximations can easily induce the numerical instability especially for hyperbolic equations and the convergence is also degraded. The regularization of the solution or the singular source terms are, thus, necessary for the spectral approximation to enhance accuracy away from the singular regions and also to maintain the stability. In [9,10] it has been shown that the direct collocation projection of the δ-function can yield spectral accuracy for some linear PDEs even though no regularization is applied. The spectral accuracy obtained by this method is due to the consistent formulation of the given PDEs in the collocation sense. The same direct projection method for the spectral Galerkin method, however, does not show such a good performance. Although, the direct collocation method is limited only to certain classes of PDEs, it can be applied to more general problems if there exist some nice properties of the solution such as symmetry, steadystate behavior, linearity, etc. [9]. The problem that we consider in this paper is described by the second-order harmonic equations with singular source terms and we will show that the direct projection method yields accurate results for this case. In this work, we perform a proof-of-concept computation to demonstrate that the spectral collocation method is more accurate and efficient compared to a time-domain solution method for the EMRI problem based on the secondorder finite-differencing in the sense of convergence. We demonstrate this using a radial infall problem for a point particle into a Schwarzschild (non-rotating) black hole. The master equation to be solved is the inhomogeneous Zerilli equation [19] which is essentially a wave-equation with a potential and a complicated source term. We solve this equation two different ways – using standard explicit second order finite difference method and using a Chebyshev spectral collocation method – and compare the results. This paper is organized as follows. In Section 2, we briefly outline the mathematics (Zerilli equation) behind a head-on collision of black-holes in the particle limit. The mass ratio of the two holes is assumed to be infinitesimally small so that the second black hole can be considered as a point source. In Section 3, we explain the numerical methodology used for the approximation of the Zerilli equation. The Chebyshev spectral collocation method is explained in detail. For the approximation of singular source terms that appear in the Zerilli equation, we introduce the direct collocation projection method of the δ-function and its derivative(s). Numerical comparison between the Chebyshev collocation method with the direct projection method and the Lax-Wendroff method for the approximation of a simple first-order wave equation is provided. In Section 4, numerical results of the Chebyshev spectral method are provided for the head-on collision case. We also present the second-order finite difference approximations for comparison with the results of the spectral method. In Section 5, we make a few conclusive comments and suggest possible future 3
work.
2
Head-on collisions of black-holes
In this section, we summarize the equations corresponding to the head-on collision of an extreme-mass-ratio binary black hole system. We will keep the discussion here very brief and simply refer the reader to the more detailed treatment presented in reference [12]. Depicted in Figure 1 is schematic that lays out the basic set-up of the problem that we are addressing in this paper. The point particle initially starts out at a location r0 and begins to “fall” toward the black hole event horizon as shown on the left side of the figure. As the particle falls, it gains more speed and radiates gravitational waves. Our numerical approach towards this problem uses the r∗ coordinate and uses appropriately chosen, finite values for the location of the boundaries.
BH Horizon
EMRI Schematic
Fig. 1. The schematic illustration of the EMRI for the 1D inhomogeneous Zerilli equation Eq. (1). R∗e and r0 denote the location of the event horizon and the initial location of the infalling point source. t denotes the physical time. As t increases, the infalling particle illustrated by the black dot with the arrow approaches the event horizon and emits the gravitational wave propagating toward R∗∞ illustrated by the solid ripples in the figure.
The central Schwarzschild black hole has a mass of M and the infalling black hole has a mass of m0. For simplicity, the mass ratio between the two holes is taken to be infinitesimally small so that the less massive hole is considered as a point source moving on a geodesic of the curved space-time due to the other, i.e. m0 /M → 0. Because we are treating this problem in the context of first-order perturbation theory, the governing equation of this system is linear and the solution is the same for any mass-ratio. One solution of a particular mass-ratio can be derived from the other of different mass-ratio, simply by scaling. We assume that the point source is falling along the z-axis toward the hole without any angular momentum. As the point source is falling into the massive hole, gravitational waves are emitted by the system. The lowest order perturbation theory of the initial Schwarzschild black hole spacetime leads to the inhomogeneous Zerilli equation with even-parity. Such an equation describes the gravitational wave ψ in 1 + 1 dimension given by the following 4
second-order wave equation with a potential and source term, ψtt = ψr∗ r∗ − Vl (r)ψ − Sl (r, t),
(1)
where r∗ is the tortoise coordinate, Vl (r) the potential term and Sl (r, t) the source term [12,19]. Note that we have set the units c = G = 1. The tortoise coordinate mentioned above, is given by r∗ = r + M ln(r/2M − 1),
(2)
where r is the physical coordinate and r > 2M. Note that tortoise coordinate r∗ approaches −∞ as the value of r approaches the Schwarzschild radius r = 2M. The Zerilli potential Vl (r) is given by Vl (r) = f(r)
2λ2 (λ + 1)r3 + 6λ2 Mr2 + 18λM 2 r + 18M 3 , r3 (λr + 3M)2
(3)
where the function f(r) is f(r) = 1 −
2M r
(4)
and the parameter λ is λ = (l + 2)(l − 1)/2.
(5)
In our case of interest, l-pole is only even with azimuthal angular term m = 0. It is noteworthy that as r → 2M, f(r) becomes f(r) → 0 and the potential vanishes. As r → ∞, the potential also vanishes. The local maximum of Vl (r) exists near the Schwarzschild radius. Now, the source term is given by 2(1 − 2M/r)κ 1 −r2 (1 − 2M/r) 0 δ 0(r − R) r(λ + 1)(λr + 3M) 2U ( ) # r(λ + 1) − M 3MU 0 r(1 − 2M/r)2 + − δ(r − R) , 2U 0 rλ + 3M
Sl (r, t) = −
where κ is q
κ = 8πm0 (2l + 1)/4π, 5
(6)
and the zero component of the 4-velocity of the particle U 0 is given by U0 =
v u u 1 − 2M/r0 t ,
1 − 2M/r
where r0 is the initial position of the particle. Singular source terms δ(r − R) and δ 0 (r − R) are due to the point source and its movement along r. Here R is the coordinate of the point source at time t. Although the source term expression takes a complicated form, one can easily obtain some intuitive aspects of the physical system from it. For example, note that the term simply scales with m0 (through the quantity defined above as κ) which is simply a reflection of our linear approximation as mentioned earlier. In addition, as expected, the expression is non-zero only at the location of the point particle (r = R). It is also interesting to note that, if the particle is very far from the black hole, it is the δ 0(r − R) term that strongly dominates the source expression, while in the other extreme case the entire term becomes negligible. The superscript 0 denotes the derivative with respect to r not r∗ . Since the derivative with respect to space in Eq. (1) is given in r∗ rather than r, we need to convert r to r∗ to locate the point source at R. For our numerical approximation in the next section with the spectral method, r∗ is transformed to the unit interval [−1, 1] using the linear map as explained in the next section. The free fall trajectory R(t) of the test particle is then given by
R˙ =
v u u 2M/R −f(R)t
− 2M/r0 . 1 − 2M/r0
(7)
If r0 → ∞, the initial system can be regarded as the unperturbed Schwarzschild spacetime and the wave function ψ vanishes for all r at t = 0. Due to the computational limit, however, r0 is only finite for the numerical approximation. For the finite r0 , the wave function ψ at t = 0 does not vanish. For this reason, the direct comparison of the numerical approximation of Eq. (1) with the analytic approach [12] with r0 → ∞ is not available. The initial approximation of ψ when r0 is finite has been investigated based on the Laplace transformation in [12]. For the current work, we assume that m0 is infinitesimally small compared to the mass M of the Schwarzschild black hole so that the initial ψ is chosen to vanish. This causes a small discrepancy in the full solution. The initial fluctuations due to such discrepancy, however, leave the computational domain according to Eq. (1) as t → ∞. Since we are interested in the decay of ψ for t → ∞, the initial fluctuations do not affect the overall decay behavior of ψ for large t. Taking m0 small, e.g. m0 = 10−10 would suffice to fulfill the main purpose of the current work. The more important initial parameter rather than m0 is the initial coordinate of the free fall, i.e. r0 = R(t = 0). Depending on r0 , the waveform of ψ with t varies. In this paper, such a parametric effect on the waveform will be considered. More a detailed study, we 6
refer the readers to [12].
3
Numerical methodology
In this section, we present the numerical methodology used in this paper - the spectral collocation method and the direct projection method of the δ-function to solve the second-order wave equation given in Eq. (1). A numerical example for the simple advection equation is also provided. For the time integration, the TVD third-order Runge-Kutta method is used for the spectral method.
3.1
Chebyshev spectral collocation method
For the spatial differentiation, the Chebyshev spectral collocation method is used. Suppose that the differential operator in one dimension is given on the solution u(x, t) for x ∈ Ω and t > 0 as Lu = 0,
(8)
with the boundary conditions B ±u = h± (t),
x ∈ ∂Ω
(9)
where B ± are the boundary operators at the domain boundary ∂Ω. Let PN be the projection operator applied to u(x, t) onto the polynomial space BN expanded by the Chebyshev polynomials of up to degree N. The Chebyshev spectral collocation method seeks a solution in the Chebyshev polynomial space expanded by the Chebyshev polynomials Tl (x) as
uN = PN u(x, t) =
m X
uˆl(t)Tl(x),
(10)
l=0
where Tl (x) is the Chebyshev polynomial of degree l and uˆl the corresponding expansion coefficient. Chebyshev polynomials are defined by the following orthogonality property
1 hl
Z1
w(x)Tl (x)Tl0 (x)dx = δll0 ,
(11)
−1
7
where hl is the normalization factor given by hl = π for l = 0 and hl = π2 for √ l ∈ Z+ and w(x) is the Chebyshev weight function given by w(x) = 1/ 1 − x2. The expansion coefficients {ˆ u l }m l=0 are found based on approximations at the collocation points defined in Ω. Here δll0 is the Kronecker delta in a usual sense. Then the exact expansion coefficients uˆl is obtained by
1 uˆl(t) = hl
Z1
w(x)u(x, t)Tl(x)dx.
(12)
−1
Since the collocation method seeks the solution at the particular collocation points, uˆl(t) is obtained by evaluating the integral using the quadrature rules on such collocation points. As long as the notation is not confusing, we also use uˆl as the approximation of the exact expansion coefficients. Since the boundary conditions are given, the commonly used collocation points are the Gauss-Lobatto quadrature points. The Chebyshev Gauss-Lobatto collocation points xj are given by
xj = − cos(
π j), N
∀j = 0, · · · , N,
(13)
here N = m for the completeness. With this condition, we construct the approximation based on the solution at the collocation points, {u(xj , t)}N j=0 such as
u(x, t) =
m X l=0
uˆl (t)Tl(x) =
m X
ψl(x)u(xl , t) =
l=0
m X
ψl (x)ul,
(14)
l=0
where let ul denote the grid function of u(x, t) at x = xj and ψl (x) are indeed interpolating polynomials. The expansion coefficients are given by
uˆl(t) =
N 1 2 X uj (t)Tl(xj ). cl N j=0 cj
(15)
The interpolating polynomial ψl (x) is a δ-function in the sense that ψl(xl0 ) = δll0 . One may adopt ψl (x) as an approximation of the δ-function, but using ψl (x) as a δ-function for the given PDE yields only highly oscillatory solution near the singularity [9]. The interpolating polynomials ψl(x) on the GaussLobatto-Chebyshev collocation points are given by
ψl(x) =
(−1)l+1 (1 − x2)TN0 (x) , cl N 2 (x − xl )
(16)
8
where cl = 1 for l = 1, · · · , N −1 and cl = 2 for l = 0, N. By taking a derivative of ψl(x) we obtain the derivative of u(x, t) with respect to x such as
u0(x, t) =
m X
ψj0 (x)uj ,
(17)
j=0
where the superscript 0 denotes the derivative with respect to x. Then at the collocation points we obtain u0 = D · u,
(18)
where u = (u0, · · · , uN )T and D the differentiation matrix given by the derivative of the interpolating polynomial at the collocation points as Dlj = ψj0 (xl ).
(19)
Thus we know that every column of D is the derivative of the δ-function at the collocation points. The elements of D with the Chebyshev Gauss-Lobatto collocation points are given by [6,8],
Dij =
ci (−1)i+j cj xi −xj
i 6= j,
xi − 12 1−x 2 i = j, i
i 6= 0, N
2N 2+1 6
i=0
2N 2+1 6
i=N
.
(20)
The Chebyshev collocation method seeks a solution u(x, t) by having the following discrete residue RN (x) vanish at the collocation points, RN (x) = LN uN − 0,
x = xj , ∀j = 0, · · · , N,
(21)
where LN = PN LPN . 3.2
Spectral projection of singular source terms
For singular source terms, we use the direct projection method [9]. The direct projection method is to approximate the δ-function in a consistent way with the given PDE. Suppose that we are given a PDE in one dimension Lu = δ(x),
x ∈ Ω = [−1, 1], 9
where L = d/dx and the boundary condition u(−1) = 0. Here the δ-function in the bounded domain Ω is defined as Z1
δ(x)dx = 1,
−1
and
Z1
δ(x)u(x)dx = u(0).
(22)
−1
Then the exact solution is given by the Heaviside function Hc (x)
Hc (x) =
0, 1,
xc
where c = 0 for this case. Then the direct projection method for the δ-function is obtained by simply mimicking the relation d Hc (x) = δ(x), dx at the collocation points such as
δN = D · H,
(23)
where δN is the collocation approximation at the collocation points, δN = (δN (x0 ), · · · , δN (xN ))T , and H is H = (0, · · · , 0, 1, · · · , 1)T . For H the nonzero value begins at x = xj such that xj > c. The consistent formula of δN does not necessarily satisfy the definition of the δ-function in Eq. (22). Suppose that u(x) = |sgn(x)|. Then by definition at each collocation point xj , d dx
Zx
δ(s)u(s)ds = δ(x)u(x) = 0,
−1
for xj , ∀j = 0, · · · , N.
At xj the consistent formula of δN yields δ(x)u(x)|xj → (D · HN )j u(xj ) = (D · H)j 6= 0, for xj , ∀j = 0, · · · , N. For the time dependent problem such as the Zerilli equation considered in the next section, the source term is also a function of time. In this case, the parameter c for Hc (x) is a function of time, i.e. c = c(t). If c(t) coincides with one of the collocation points, xj at a certain time tp, the definition of the Heaviside function Hc (x) at xj and tp can be either Hc(tp ) (xj ) = 1 or Hc(tp ) (xj ) = 21 . Both definitions yield the same results. 10
3.3
Numerical examples
To demonstrate the accuracy obtained by the direct projection method for the 1D wave equation with the singular source term, we consider the following advection equation ut + ux = δ(x),
x ∈ [−1, 1],
t > 0,
(24)
with the initial condition u(x, 0) = − sin(πx) and u(−1, t) = − sin(π(−1 − t)) for t > 0. The exact solution of Eq. (24) is then given by u(x, t) = − sin(π(x − t)) +
Zx
δ(x)dx.
−1
As the exact solution implies, the solution u(x, t) is discontinuous at x = 0. The jump magnitude [u] at x = 0 is 1 for any time t, +
−
[u]|x=0 := u(0 ) − u(0 ) =
Z0+
−1
δ(x)dx −
Z0−
δ(x)dx = 1.
−1
For the numerical experiment, we consider N = odd. Then the exact solution u(x, t) at the Gauss-Lobatto collocation points x = xi and at t are given by u(xi , t) =
− sin(π(xi − t)) i ∈ [0, · · · , N2+1 ]
1 − sin(π(xi − t))
i ∈ [ N2+3 , · · · , N]
,
with x(N +1)/2 < 0 < x(N +3)/2. Let v be the spectral approximation to Eq. (24) such that v = (v0, · · · , vN )T . Here vj denotes the grid function at x = xj . Then we seek v with the direct projection method such that dv + D · v = D · H, dt
(25)
where dtd denotes the time integration operator applied to the discrete v. For the time integration, we use the 3rd order TVD Runge-Kutta scheme [14]. The boundary condition is directly applied v0 = − sin(π(x0 − t)),
(26)
after the final stage of the RK step at the given time t. 11
Suppose that the numerical approximation of v(x, t) is given in the form of v(x, t) = w(x − t) + z(x) where the vectors w(x − t) and z(x) are the timedependent and time-independent parts of v(x, t) respectively. With the method of line, Eq. (25) becomes dw + D · (w + z) = D · H, dt and it is clear that dw + D · w = 0, dt
D · z = D · H.
The boundary condition is given by v(−1, t) = u(−1, t) = − sin(π(−1 − t)) +
Zx
−1
δ(x)dx = − sin(π(x − t)),
that is, w(−1, t) = − sin(π(−1−t)) and z(−1) = 0. Thus we know that w(x, t) is the spectral collocation approximation of wt + wx = 0 because its exact solution is given by w(x, t) = − sin(π(x − t)). The time-independent solution z(x) is then obtained from by solving D · z = D · H. Since the boundary value of z(x) is given as z(−1) = 0, z(x) is given by ˜ · (˜ ˜ = 0, D z − H) where the superscript ˜ denotes the inner matrix or vector without the first ˜ is non-singular. Thus z(x) = H(x) column and row. D is singular, but D exactly. Here note that such exactness is obtained only at the collocation points. Due to the uniqueness of the solution of Eq. (25), v(x, t) = w(x, t)+z(x) indeed. The error function E(x, t) is given by E(x, t) = u(x, t) − v(x, t) = − sin(π(x − t)) + H(x) − (w(x, t) + z(x)) = − sin(π(x − t)) − w(x, t). Thus we know that the error is solely determined by the inhomogeneous solution w(x, t) which is smooth function for ∀t > 0. Thus, the error decays spectrally. Since the initial condition v(x, 0) is simply given by v(x, 0) = − sin(π(x − t)), there exist the Gibbs oscillations due to the initial discrepancy of the initial condition. To see how the initial discrepancy changes with time, t > 0, let e(x, t) = v(x, t)−v e(x, t) where v e (x, t) is the numerical solution with the exact initial condition. By definition, e(x, 0) 6= 0. Since the boundary condition is the same for both v and v e , we know that e(−1, t) = 0 for ∀t > 0. Since both v and v e are obtained from the same equation, we obtain de + D · e = 0. dt Since e(−1, t) = 0 for ∀t ≥ 0, e(x, t) = 0 for t > 2. Thus the initial Gibbs oscillations due to the discrepancy of the initial condition do not affect the solution behavior after a finite time t. 12
Finite difference scheme: For the comparison, we consider the Lax-Wendroff scheme to solve Eq. (24) with the direct projection method. Suppose that the finite difference solution is sought with the evenly spaced grids for a given N such that xj = x0 + jh, x0 = −1, h ≡ ∆x = 2/N.
Let D+ , D− and D0 the forward, backward and central difference operators defined as [7] hD+ vj = vj+1 − vj ,
hD− vj = vj − vj−1 ,
2hD0 vj = vj+1 − vj−1.
We also use the direct projection method for the approximation of the δfunction as dHN (x) δN (x) = , dx d where dx is now the central difference operator D0 with the Lax-Wendroff scheme such that dHN (x) HN (xj+1 ) − HN (xj−1 ) → D0 HN (x) = , dx 2h
(27)
where HN (x) is the grid function with the finite differencing grid of H0 (x). This definition of δN (x) satisfies the definition of the δ-function almost everywhere in the domain. For u(x) = |sgn(x)|, δN (x) yields for even N, d dx
Zx
−1
δ(s)u(s)ds → (D · HN )j u(xj ) = (D · H)j =
Then the Lax-Wendroff scheme is given by vjn+1
=
vjn
+
dtD0 vjn
1 , 2h
xj = −∆x, ∆x
0,
otherwise
.
dt2 + − n + D D vj + D0 HN , 2
where dt is the time step and we choose dt as small as the stability is guaranteed. Gaussian regularization: For the numerical comparison, we also consider the Gaussian type regularization of the δ-function given as 1 δN (x) = √ exp(−x2/22 ), 2π where is the free parameter for the regularization. We choose = 2/N for the numerical experiments. 13
To measure the errors we define the discrete L2 and L∞ errors as L2 Error =
v u u t
N 1 X (v(xi, t) − u(xi, t))2, N + 1 i=0
and L∞ Error = max |v(xi, t) − u(xi, t)|. i∈[0,···,N ]
Figure 2 shows the approximations δN of δ(x) with the spectral method given in Eq. (23) with N = 65 (left) and the centered difference scheme given in Eq. (27) with N = 66 (right). As shown in the figures, the direct projection approximation of δ(x) with the Chebyshev method is highly oscillatory over the whole domain. Note again that N is assumed to be odd to avoid any ambiguity at x = c = 0 for the spectral method. The approximation with the centered difference method, however, yields the approximation without any oscillations unlike the spectral approximation. The figure also shows that the δ-function is highly localized around the singularity at x = 0 with the finite difference method. The oscillations shown in the spectral approximation is simply, the Gibbs phenomenon. It is also well known that the spectral solution is highly oscillatory if there is a local jump discontinuity such as the δ-function. As mentioned earlier, in [9] it has been shown that the direct projection method yields an accurate solution and spectral accuracy can be also achieved for certain types of PDEs although the approximation of the δ-function obtained by this method is highly oscillatory. 15 15
10
N
δ (x)
5
N
δ (x)
10
0 5 −5 −10
0 −1
−0.5
0
0.5
1
−1
x
−0.5
0
0.5
1
x
Fig. 2. Left: The direct spectral projection approximation of δ(x) on the Chebyshev Gauss-Lobatto collocation points. Right: The direct central difference approximation of δ(x) on the evenly spaced grids with the central difference operator D 0 for N = 66.
Figure 3 shows the solutions and pointwise errors at t = 10. The left figure of Figure 3 shows the solutions with the Chebyshev spectral (top) and Lax-Wendroff (bottom) methods. The red and blue lines represent the exact solution and the approximation, respectively. For the spectral approximation, the exact solution and the approximation are not distinguishable. There are no Gibbs oscillations observed in the spectral approximation. It seems that 14
the direct spectral approximation of the δ-function yields a cancellation of these oscillations. In [9], it has been discussed that such cancellation is due to the consistent formulation to the given PDE. For the approximation with the Lax-Wendroff method, small oscillations are seen around the singularity at x = 0. In the right figure of Figure 3, the pointwise errors versus x are given. The figure shows that the Chebyshev spectral method yields very accurate result over the whole domain without any indication of the degradation of convergence order. Indeed, with the Chebyshev spectral method, the spectral accuracy and uniform convergence are recovered even with the local jump discontinuity. Contrary to the spectral approximation, the approximation with the Lax-Wendroff method only shows the low order accuracy. 0 −2
0.5
Log10 Error
u(x,t), v(x,t)
1
0 −1
−0.5
0
0.5
1
x u(x,t), v(x,t)
1
−4 Lax−Wendroff
−6 −8 −10 −12
0.5
Chebyshev
−14 0 −1
−0.5
0
0.5
−16
1
x
−1
−0.5
0
0.5
1
x
Fig. 3. Left: Solutions at t = 10 with the Chebyshev spectral (top) and Lax-Wendroff (bottom) methods. The red and blue lines represent the exact and the numerical solution, respectively. Note that the exact and numerical solutions are not distinguishable for the Chebyshev spectral method. Right: Pointwise errors Log10 |u(x, t) − v(x, t)| versus x with the Lax-Wendroff method and Chebyshev spectral methods.
Figure 4 shows the trace of approximations with time with the Chebyshev spectral method (left) and the Lax-Wendroff method (right). There are small oscillations in the beginning time stages with both methods. These oscillations are due to the inconsistent initial condition. These oscillations with the Chebyshev spectral method, however, leave the domain quickly as time goes on while they remain with the Lax-Wendroff method. The right figure for the Lax-Wendroff method clearly shows that such oscillations exist with all time t ∈ (0, 5] while no significant oscillations are seen in the left figure for the Chebyshev spectral method even after t > T . Here T is the time for the initial Gibbs oscillations to leave the domain and it is about T ∼ 1 as the wave speed is 1. Table 1 shows the L2 and L∞ errors of the solution for various approximations of the δ-function. In the table, LW-Direct, LW-Gaussian, SP-Direct and SPGaussian represent the Lax-Wendroff method with the direct and Gaussian approximations of the δ-function, and the Chebyshev spectral method with the direct and Gaussian approximations of the δ-function, respectively. As shown 15
Fig. 4. Trace of solutions with time with the Chebyshev spectral (left) and Lax-Wendroff methods (right).
in the table, the best result is obtained with the Chebyshev spectral method with the direct approximation of the δ-function. The table also shows that the direct projection method yields a slight better result for the Lax-Wendroff method. Table 1 L2 and L∞ errors with various approximations of δ(x) at t = 10 for Eq. (24). Method
L2 error
L∞ error
LW-Direct
1.02(-2)
2.52(-2)
LW-Gaussian
4.52(-2)
2.58(-1)
3.60(-10)
7.77(-10)
4.04(-2)
2.33(-1)
SP-Direct SP-Gaussian (n) = 10n
4
Numerical approximation of Zerilli equation
Now we consider numerical approximations of the inhomogeneous Zerilli equation with the spectral and finite difference methods. For both the spectral and finite difference methods, we use the direct projection method of the δfunction. Here we note that the finite difference method used in this work is the explicit second-order centered finite difference method. There have been developed high order finite difference methods for the Zerilli equations such as the fourth-order convergent finite difference method [11]. The comparisons between the spectral and high order finite difference approximations for the Zerilli equations will be left for our future work. The governing equation is defined on the simple 1+ 1 spacetime with time and space denoted by t and r∗ respectively. The range of r∗ , the tortoise coordinate 16
is given by R∗e ≤ r∗ ≤ R∗∞ ,
where R∗e is obtained close to the radius of the event horizon and R∗∞ → ∞. In practice, R∗e and R∗∞ can not be −∞ and ∞ respectively. Let R∗e is taken at r ∼ 2M and R∗∞ at r 2M. By definition, R∗e < 0 and R∗∞ > 0. The tortoise coordinate r∗ is linearly mapped into the Chebyshev Gauss-Lobatto collocation points {ξj }N j=0 given as r∗ =
R∗ + R∗e R∗∞ − R∗e ξ+ ∞ , 2 2
(28)
where ξj = − cos(πj/N), 4.1
j = 0, · · · , N.
Derivative operator D
Using the linear mapping of Eq. (28), the derivative operator with respect to r∗ , D is given by D :=
∂ 2 ∂ 2 = = D(ξ), ∂r∗ R∗∞ − R∗e ∂ξ R∗∞ − R∗e
where D(ξ) is the differential operator defined on the Gauss-Lobatto collocation points {ξj }N j=0 given in Eq. (20). The second derivative operator is given by !2 2 2 D =D·D = D2 (ξ). R∗∞ − R∗e Then the Chebyshev spectral method for the head-on collision system Eq. (1) seeks an approximation for d2 ψ = D2 ψ − Vl ψ − S, dt2
(29)
where ψ, Vl and S are the wave function, the potential of l mode and the source term defined on {ξj }N j=0 , respectively. Singular source terms δ(r, t) and 0 δ (r, t) are approximated with the direct projection method as δ(r − R) = D · HR ,
δ 0 (r − R) = D · δ(r − R) = D2 · HR ,
where H is the Heaviside function defined on {ξj }N j=0 . Here we note that R is a function function of time and the Heaviside function HR is being shifted as time goes on. 17
In our computation, the second derivative of ψ is with respect to r∗ not r. Thus at each collocation point, r∗ , the corresponding proper distance should be calculated to obtain the potential and singular source terms accordingly, for which the bi-section method is used.
4.2
Time stepping
For the time-stepping, we adopt the second-order time stepping method ψtt =
ψ(t + ∆t) − 2ψ(t) + ψ(t − ∆t) . ∆t2
This method is two-step method. Thus we need two initial conditions for the time marching. We can simply use ψ(r, 0) = ψ(r, ∆t), or we can evolve Eq. (1) using the first order approximation for ψ(r, ∆t) [7].
4.3
Initial condition
The initial location of the point particle is r = r0 < ∞. We have to find the exact initial condition when the particle is located at finite r = r0 initially. In this paper, we assume that ψ(r, t = 0) = 0. This vanishing initial condition is not physical and inconsistent to Eq. (1) as it does not satisfy the governing equation. Due to the inconsistent treatment of the initial condition, the numerical approximations of ψ in the beginning stages are highly oscillatory and it takes a certain time for these initial oscillations to leave the computational domain as we will see in the next sections.
4.4
Boundary conditions
We impose the absorbing boundary condition at the boundaries r∗ = R∗e and r∗ = R∗∞ . Since this is the 1D problem, the following boundary condition should be a perfectly absorbing boundary condition: at r∗ = R∗e ,
ψt − ψr∗ = 0, 18
and at r∗ = R∗∞ .
ψt + ψr∗ = 0,
Here we note that the derivative with the spectral method at the end points should not be obtained using the global derivative operator D because there can be a discontinuity inside the domain. That is, the first order derivative term is not given as D · ψ|r=Re . The no-flux boundary conditions should be only local. We use the upwind method at both ends such as ψ(r1∗ , t) − ψ(r0∗ , t) ψ(r0∗, t + ∆t) − ψ(r0∗, t) = , ∆t r1∗ − r0∗
at r∗ = R∗e ,
where r0∗ = R∗e and ∗ ∗ ψ(r0∗, t + ∆t) − ψ(r0∗, t) ψ(rN , t) − ψ(rN −1 , t) =− , ∗ ∗ ∆t rN − rN −1
at r∗ = R∗∞ ,
where r0∗ = R∗∞ .
4.5
Spectral filtering methods
We also adopt the spectral filtering method to minimize the possible nonphysical high modes. The oscillations with the spectral method possibly found near the local jump discontinuity and also generated due to the inconsistent initial conditions, propagate through the whole domain. To reduce the nonphysical high modes, the filter function is applied to the solution. To apply the filter function, we transform the approximation found in the physical domain to the coefficient domain using Eq. (15). Then we obtain the filtered approximation as uσN (xl , t) =
N X
σ(j/N)ˆ uj Tj (xl),
j=0
where σ(l/N) is the filter function according to [20]. The filtering method is equivalent to the spectral viscosity method [8]. For the current work, we use the exponential filter defined as σ(j/N) = exp(ln (j/N)p ),
∀j = 0, · · · , N,
where is a strictly positive real constant and p is known as the filtering order. The filter function is constructed to satisfy the properties, σ(0) = 1, σ (q)(0) = 0 for q = 1, · · · , p − 1, σ(±1) = 0, and σ(−η) = σ(η) [20]. In principle, uσN (t) → uN (t) if p → ∞. If p → 0+ , uσN (t) becomes only 0th order approximation of uN (t). By the definition of the filter function σ(j/N), σ(1) → 0, for which the positive constant is usually chosen so that exp(ln ) becomes about machine accuracy, M . For our work, we choose M ∼ 10−16 . 19
Using Eq. (15), we have uσN (xi )
=
N X
Sij uj ,
j=0
∀i = 0, · · · , N,
where Sij are Sij =
4.6
N 2 1 X Tn (xi)Tn (xj ) exp(ln (n/N)p ). cj n=0 cn N
Filtering for the Lax-Wendroff scheme
For the Lax-Wendroff scheme, we also experimented by applying a filtering method. We simply convolved a Gaussian distribution with the source-term itself, yielding a smoother profile for it. This Gaussian filter approach that has been used elsewhere [21] in literatures is a natural one to attempt in our context. Through experimentation, we realized that choosing the width of the Gaussian to be of the same scale as the spacing between grid points yields the best results. Figures 8,17, and 18 depict sample results from performing such Gaussian filtering in the finite difference evolutions. These results show that the Gaussian filtering yield better performance than those without the filtering. We will, however, show that the filtering method does not completely remove the oscillations for large t resulting in an inaccurate approximation of the ringdown of the gravitational wave. With the Gaussian convolution the filtered function is given by fNσ (x, t)
=
Z∞
−∞
f(x, t) exp(−(x − y)2/σ)dy,
The Gaussian filter is applied for the source term. Since the source is only localized, the integration is done only over a certain number of grids, that is, for example at ith grid point, x = xi , the integration on the grids is done over [xi−np , xi+np ]. Thus totally 2np + 1 grid points are used for the integration. According to our numerical experiments, the best parameters for the Gaussian kernel for various N are (N, np , σ) = (32, 2, 8), (64, 6, 8), (128, 32, 16), and (1024, 32, 32). These values are used for the filtering for the finite difference method in the following sections. 4.7
Numerical results
For the numerical experiments, we use M = 1, m0 = 10−10 , R∗e = −30, R∗∞ = 30, 130, and l = 2. For the time integration we use dt = 5 × 10−5 with 20
the final time Tf inal = 300, 600, and 2400. For the spectral collocation method, the range of the filtering order has been chosen such that the overall power-law decay of the waveform is ensured. According to our numerical experiments, the filtering orders in the range of p = [32, 64] have been used. For the finite difference method, the filtering parameters given in Section 4.6 are used. Since three different waveforms and spectra are obtained depending on the value of r0 /2M, we use r0 /2M ∼ 2, r0 /2M ∼ 2 to 5 and r0/2M > 5. In [12], it has been discussed about the evenly spaced bumps when r0 /2M > 5. In order to verify such evenly spaced bumps, we also use r0 /2M = 50. For the comparison of the spectral method with the finite difference method, we focus on accuracy and convergence of the ringdown profile with time and N. Since the current work is based on the spectral method with the single domain, the finite difference method has faster computing time than the spectral method. The multi-domain spectral method will be considered in our future work to increase the computing speed.
4.7.1 r0 = 1.5(2M) First we consider the case of r0 = 1.5(2M) for which the waveforms are given as a simple quasinormal ringing. For the numerical experiments, R∗e = −30, and R∗∞ = 130 are used. Figure 5 shows the waveform, ψ/m0 as a function of time at r = R∗∞ . At t = 0, we use ψ = 0. The left figure shows the results with the Chebyshev spectral collocation method with N = 128 and the right figure with the centered finite difference method with N = 128 and N = 1024. Both results are obtained without filtering. The results with both the spectral and finite difference methods show the ringdown profile around t ≥ 150. The waveform is high oscillatory and it is not decaying with the finite difference method for t > 150 while the spectral method shows the properly decaying waveform. Figure 6 shows the filtered waveforms with
0.6
0.4
0.4
0.2
0.2
ψ/m
ψ/m
0
0.8
0.6
0
0.8
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8 0
−0.8 0
50
100
150
200
250
300
t
N = 128 N = 1024 50
100
150
200
250
300
t
Fig. 5. ψ/m0 versus t. Left: The spectral method with N = 128. Right: The 2nd order finite difference method with N = 128 (red) and N = 1024 (black).
different filtering orders such as p = 64 (top), 48 (middle), and 32 (bottom). 21
The left figure shows the results when the filter on-set time, tf ilter is tf ilter = 0 (left) and tf ilter = 10 (right). It is interesting that the initial oscillations due to the inconsistency of the initial condition are gradually amplified if tf ilter = 0, but they are amplified instantly at t = tf ilter with tf ilter = 10. As the figures show, the filter function moderates the solution and yields non-oscillatory ringdown waveforms for t ≥ 150 while it yields oscillatory waveforms for t ≤ 130. These oscillations originate from the inconsistent initial condition at t = 0 and are amplified with the filter applied. Since ψ(·, 0) → 0 only as r0 → ∞, the inconsistent initial condition induces the initial oscillations. These oscillations propagate over the domain and eventually leave the domain. As time goes on, the solution seems to become more consistent to the given equations and the numerical solution becomes more regulated. The spectral solution without the filter shows a small oscillations until the initial oscillations leave the domain. The spectral solution with the filter, however, shows that the initial oscillations are amplified with the filter until the solution reaches the ringdown phase at around t ∼ 130. For t > 130, the oscillations are reduced significantly in the filtered spectral solutions. The numerical results with other N and p show the similar amplification effects.
ψ/m
ψ/m
0
0.5
0
0.5 0 −0.5 0
50
100
150
200
250
−0.5 0
300
100
150
200
250
300
50
100
150
200
250
300
50
100
150
200
250
300
0
ψ/m
ψ/m
50
0.5
0
0.5 0 −0.5 0
50
100
150
200
250
0 −0.5 0
300
ψ/m
0
0.5
0
0.5
ψ/m
0
0 −0.5 0
50
100
150
200
250
0 −0.5 0
300
t
t
Fig. 6. ψ/m0 versus t with the spectral method for p = 64 (top/red), p = 48 (middle/blue) and p = 32 (bottom/magnet). Left: tf ilter = 0. Right: tf ilter = 10.
In order to check the filtering effect at different locations, the values of ψ/m0 with time are collected at different locations for N = 128. Figure 7 shows the results. The left top and bottom figures show ψ/m0 at r∗ = 127.6025 and r∗ = 120.5537 and the right top and bottom at r∗ = 94.4456 and r∗ = 57.8414, respectively. Note that Figure 6 is obtained at r∗ = 130. Also note that these locations are 10, 20, 40, and 60 grid points away from the grid at r∗ = 130, respectively. As shown in the figure, the amplification effects by the spectral filtering are found at every grid point. Such unusual amplification effect of the inconsistent initial condition by the spectral filtering method is not explained in the current work. We will investigate the amplification effect by the filtering method and explain its mathematical reason in the future work. 22
0.5
ψ/m0
ψ/m0
0.5
0
−0.5 0
50
100
150
200
250
−0.5 0
300
50
100
150
200
250
300
50
100
150
200
250
300
0.5
ψ/m0
ψ/m0
0.5
0
−0.5 0
0
50
100
150
200
250
0
−0.5 0
300
t
t
Fig. 7. ψ/m0 versus t with N = 128 and p = 64 at r ∗ = 127.6025 (left top), 120.5537 (left bottom), 94.4456 (right top), and 57.8414 (right bottom).
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
ψ/m0
ψ/m0
Figure 8 shows ψ/m0 with the finite difference method with the Gaussian filtering. As the figure shows the Gaussian filtering significantly reduces the oscillations found in Figure 5. Moreover, the amplification effects by the filtering are not found with the finite difference method. In Figure 18, however, we will show that the spectral method is more accurate and has faster convergence than the finite difference method.
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8 0
50
100
150
200
250
300
t
−0.8 0
50
100
150
200
250
300
t
Fig. 8. ψ/m0 versus t with the finite difference method with the Gaussian filtering. Left: N = 128. Right: N = 1024.
Figure 9 shows the waveform ψ/m0 of unfiltered and filtered spectral solutions for 130 ≤ t ≤ 180 with N = 128. For filtered solutions, p = 64, 48, and 32 are used. The left figure shows the case of tf ilter = 0 and the right of tf ilter = 10. After the solution becomes more regularized, the oscillations shown in the unfiltered solution are considerably reduced in the filtered solutions, and the filtered solutions with different p agree well. It is also observed that the filter on-set time does not affect the solution in the ringdown phase if tf ilter ≤ t ∼ 130. Figure 10 shows the waveforms ψ/m0 of filtered spectral solutions and those of the finite difference method. N = 128 is used for the spectral method and N = 128 (left) and N = 1024 (right) are used for the finite difference method. No filtering Gaussian filtering is used for the finite difference method. The figures show that the waveforms are highly oscillatory with the finite difference method and also show that more oscillatory waveforms are found with 23
larger N. Furthermore the figures show that the global behavior of the finite difference solution with large N = 1024 agree with that of the filtered spectral solutions better than with N = 128. This implies that the spectral method can yield more accurate solution with smaller N than the finite difference method with the proper choice of p. 0.8
0.8
Non−Filtered Solution
0.6
0.4
0.2
0.2
ψ/m
0
0.4
0
ψ/m
Non−Filtered Solution
0.6
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8 130
140
150
160
170
−0.8 130
180
140
150
t
160
170
180
t
Fig. 9. ψ/m0 versus t with the spectral method for p = 64 (red), p = 48 (blue), p = 32 (magnet) and unfiltered method (black) for t ∈ [130, 180]. Left: tf ilter = 0. Right: tf ilter = 10. 0.8
0.8
0.6 0.4
0.6
}
SP with p = 64, 48, 32
0.4
ψ/m
ψ/m
0
0.2
0
0.2 0 −0.2
−0.2
Finite Difference with N = 128
−0.4
Finite Difference with N = 1024
−0.4
−0.6 −0.8 130
0
−0.6 140
150
160
170
−0.8 130
180
t
140
150
160
170
180
t
Fig. 10. ψ/m0 versus t with the spectral and finite difference methods. For the spectral method, p = 64 (red), p = 48 (blue), and p = 32 (magnet) are used with N = 128 and tf ilter = 0 for both plots. For the finite difference method, N = 128 (left) and N = 1024 (right) are used.
Based on the solutions obtained, we conduct a Fourier analysis to find the period T and the frequency ω of the ringdown waveforms. Figure 11 shows the power spectrum of the waveform ψ/m0. Since there still exists the initial non-physical oscillations in the waveform until t ∼ 130, we use the waveforms for t ≥ 131.1 after the solution is regularized. The final time is t = 150. Figure 11 shows the power spectrum of ψ/m0 obtained by the filtered spectral method with N = 128 and p = 32 (left) and the finite difference method with N = 1024 (right). The fundamental frequency is clearly seen in both the left and right figures. Since the finite difference solution with N = 1024 is high oscillatory as shown in Figure 10, there are secondary dominant frequencies found in the 24
power spectrum. Table 2 shows the period and frequency obtained with the Fourier analysis. In the table CD denotes the finite difference method with the central differencing, SP the spectral method and p the spectral filtering orders. The table shows two sets of period and frequency, one from the power spectrum using the data for 131.1 ≤ t and the other for 131.9 ≤ t. The period computed based on the frequency domain approach is well known as T ∼ 16.8. The table clearly shows that the periods of the ringdown waveforms with the spectral method for N = 32 to N = 128 and with the finite difference method for N = 128 and N = 1024 are about T ∼ 16.8. The finite difference method with smaller N such as N = 32, 48, and 64, however, yields inaccurate results of T ∼ 18.7 and T ∼ 15.3. Both the spectral and finite difference methods with N = 128 yield almost the same period and frequency although they provide different waveforms as shown in Figure 10. The table also shows that the spectral method yields an accurate period even with smaller N such as N = 32.
10
Power
10
10
10
10
10
10
−2
10
−3
10
−4
10
Power
10
−5
−6
−7
10
10
10
−8
10
−9
0
2
4
6
8
10
10
12
Frequency
−2
−3
−4
−5
−6
−7
−8
−9
0
2
4
6
8
10
12
Frequency
Fig. 11. Power spectrum. Left: With the spectral method with p = 32 and N = 128. Right: With the centered finite difference method with N = 1024.
We also compute the waveforms with different N. Figure 12 shows the waveforms with the spectral method (left) and the finite difference method (right). No filtering method is applied for both methods. From top to bottom, each figure shows the waveform for N = 64, 48, and 32. The waveforms with the finite difference methods show the oscillatory behavior even after 130 ≤ t and they are not decaying. The figures with the spectral method show that the method yields a non-oscillatory decaying ringdown profile with small N such as N = 48. Figure 13 shows the waveforms with R∗∞ = 30 and r0 = 1.5(2M). The left figure shows the unfiltered spectral solutions for N = 128 (top), N = 64 (middle), and N = 32 (bottom). The right figure shows the filtered spectral solutions for N = 128 with the filtering orders p = 64 (top), p = 48 (middle), and p = 32 (bottom). As the figure shows the ringdown waveforms reach the computational boundary at an earlier time than with R∗∞ = 130. Consequently the initial oscillations seem to exist in the ringdown profiles. The quasinormal 25
Table 2 The period T and frequency ω of the ringdown of the waveform ψ/m0 for l = 2 and r0 = 1.5. The time intervals are I1 = [131.1, 150] and I2 = [131.9, 150]. Method
N
CD
p
ω
T (I2 )
ω
T (I1 )
32
0.33649906435356
18.672
0.33490475430629
18.761
CD
48
0.33649906435356
18.672
0.33490475430629
18.761
CD
64
0.41127663420991
15.277
0.40932803304102
15.350
CD
128
0.37388784928174
16.805
0.37211639367365
16.885
CD
1024
0.37388784928174
16.805
0.37211639367365
16.885
SP
32
0.37388784928174
16.805
0.37211639367365
16.885
SP
48
0.37388784928174
16.805
0.37211639367365
16.885
SP
64
0.37388784928174
16.805
0.37211639367365
16.885
SP
128
0.37717970828492
16.658
0.37717970828492
16.658
SP
128
64
0.37388784928174
16.805
0.37211639367365
16.885
SP
128
48
0.37388784928174
16.805
0.37211639367365
16.885
SP
128
32
0.37388784928174
16.805
0.37211639367365
16.885
0.5 0
ψ/m
ψ/m
0
0.5 0 −0.5 0
50
100
150
200
250
−0.5 0
300
100
150
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250
300
50
100
150
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300
50
100
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300
0
ψ/m
ψ/m
50
0.5
0
0.5 0 −0.5 0
50
100
150
200
250
0 −0.5 0
300
ψ/m
0
0.5
0
0.5
ψ/m
0
0 −0.5 0
50
100
150
200
250
0 −0.5 0
300
t
t
Fig. 12. The waveforms ψ/m0 with N = 64(top), 48(middle), and 32(bottom). Left: Spectral method without filtering. Right: Finite difference method without filtering.
waveforms are, however, still well observed as the case of R∗∞ = 130. When r0 is as small as r0 < 5M, the waveforms yield the similar quasinormal forms as those shown in Figures 6 and 12.
4.7.2 r0 ≥ 5(2M) We now consider the case of r0 ≥ 5(2M). For the numerical experiments, we use r0 = 5(2M) and r0 = 10(2M). Figure 14 shows the waveforms ψ/m0 versus t with the spectral method for N = 128 and r0 = 5(2M) (left) and 26
ψ/m
ψ/m
0
1
0
1 0 −1 0
50
100
150
200
250
300
100
150
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300
50
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50
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300
0
ψ/m
ψ/m
50
1
0
1 0 −1 0
50
100
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250
0 −1 0
300
ψ/m
0
1
0
1
ψ/m
0 −1 0
0 −1 0
50
100
150
200
250
0 −1 0
300
t
t
Fig. 13. ψ/m0 versus t with R∗∞ = 30, and r0 = 1.5(2M ). Left: Unfiltered solution. N = 128(top), 64(middle), and 48(bottom). Right: Filtered solution for N = 128 with p = 64, 48, and 32. .
r0 = 10(2M) (right). From top to bottom, each figure shows the waveforms with the unfiltered spectral method, and filtered spectral method with p = 64, p = 48 and p = 32, respectively. The figures show that the waveforms for r0 ≥ 5(2M) are different from those obtained with r0 = 1.5(2M). The overall waveforms for r0 = 5(2M) and r0 = 10(2M) are similar. One major difference of the waveforms between the cases of r0 = 1.5(2M) and r0 = 5(2M), 10(2M) is that there exists a series of bumps before the ringdown of the waveforms starts. The weveforms for r0 = 10(2M) show a longer series of such bumps than that of r0 = 5(2M). In [12], the series of bumps are briefly explained and it has been shown that these bumps are evenly spaced with t. 2 0
ψ/m
ψ/m
0
2 0 100
150
200
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300
ψ/m
ψ/m
0
150
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0
50
ψ/m
ψ/m
100
0 −2 0 2
0
−2 0 2 0
100
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300
ψ/m
0
50
0 −2 0 2
0
−2 0 2
ψ/m
50
0
50
0 −2 0 2
0
−2 0 2
0 −2 0
50
100
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250
0 −2 0
300
t
t
Fig. 14. ψ/m0 versus t with the spectral method for the unfiltered solution (black), p = 64 (red), p = 48 (blue), and p = 32 (magnet) from top to bottom for N = 128. Left: r0 = 5. Right: r0 = 10.
4.7.3 r0 = 50(2M) In order to look more closely at the series bumps of the waveforms for large r0 , we compute the waveform for r0 = 50(2M). Figure 15 shows the results with the spectral and the finite difference methods. The left figure of Figure 27
15 shows the unfiltered spectral approximations for N = 128, 64, 48, and 32 from top to bottom. These spectral approximations look similar to each other, but the figures clearly show that the number of bumps found around the interval t ∈ [200, 1200] decreases as N decreases. The locations of bumps almost coincide for all N and more bumps are found with higher N between these coinciding bumps. The gravitational ringdowns are observed after t > 1200 for all N and it is observed that these ringdown profiles are decaying as t goes on. The right figure of Figure 15 shows the approximation with the finite difference method for N = 1024, 128, 64, and 32. The figures show that the bumpy structures in the waveform are not clearly seen with the finite difference method compared to the spectral method although the overall waveforms are very similar. In the ringdown phase, all the spectral methods yield the decaying waveforms but the waveforms with the finite difference method are very oscillatory and do not decay.
2 0 −2 −4 −6
0
ψ/m 1000
1500
2000
ψ/m
0
500
0
500
1000
1500
2000 0
2 0 −2 −4 −6
0
ψ/m
2 0 −2 −4 −6
0
500
1000
1500
2000 0
2 0 −2 −4 −6
ψ/m
ψ/m
0
ψ/m
0
ψ/m
0
ψ/m
0
Figure 16 shows the waveforms with time in logarithmic scale for t ∈ [0.0316, 31.6228] (top left), t ∈ [31.6228, 316.2278] (top right), t ∈ [316.2278, 630.9573] (bottom left), and t ∈ [1584.893, 2000] (bottom right). In each figure, the black, red and blue solid lines represent the unfiltered spectral approximations with N = 128, N = 64, and the filtered spectral approximation with N = 128 and p = 64, respectively. The magnet dotted line represents the approximation with the finite difference method with N = 128. There is a difference of the waveform in the beginning stages among the unfiltered and filtered spectral approximations. The waveforms of filtered and unfiltered spectral approximations, however, begin to agree well for large time t. Especially the bottom right figure shows that the spectral solutions yield almost the same decaying waveforms for N = 128 and N = 64. For very large t > 1500, the unfiltered finite difference method does not show the proper decaying behavior of the waveform as shown in the figure.
0
500
1000
1500
2000
t
2 0 −2 −4 −6 2 0 −2 −4 −6 2 0 −2 −4 −6 2 0 −2 −4 −6
0
500
1000
1500
2000
0
500
1000
1500
2000
0
500
1000
1500
2000
0
500
1000
1500
2000
t
Fig. 15. ψ/m0 versus t. Left: Spectral approximations with N = 128, 64, 48, and 32 from top to bottom. Right: Finite difference approximations with N = 1024, 128, 64, 32.
Figure 17 shows the same plots with the finite difference method as in Figure 16 28
−2
0.6 0
Log |ψ/m |
0.8
0
Log |ψ/m |
0
10
10
−4 −6
0.4
0.2 −8 −1.5
−1
−0.5
0
0.5
1
0 1.5
1.5
1.7
Log10 t
2
2.3
2.5
Log10 t −3 −4 0 10
0.3
Log |ψ/m |
0
0.4
10
Log |ψ/m |
0.5
0.2 0.1 0 2.5
−5 −6 −7 −8 −9
2.55
2.6
2.65
2.7
2.75
−10 3.2
2.8
Log10 t
3.25
3.3
3.35
3.4
Log10 t
Fig. 16. Log10 |ψ/m0| versus Log10 t with different time intervals. Top left: t ∈ [0.0316, 31.6228]. Top right: t ∈ [31.6228, 316.2278]. Bottom left: t ∈ [316.2278, 630.9573]. Bottom right: t ∈ [1584.893, 2000]. The dotted line represents the approximation with the finite difference method with N = 128. The black, red and blue solid lines represent the spectral method with N = 128, N = 64 and N = 128 with p = 64, respectively.
for N = 1024, 128, 64, and 32. For the figures, the Gaussian filtering has been used. We observed that the power-law decay of the ringdown profile is obtained with N = 1024 with the proper Gaussian filtering method. However, it is also observed that the decay profiles for large t are still contaminated with small oscillations. Here note that the results in Figure 16 with the spectral method are obtained without the filtering method. The power-law decay profile with the finite difference method with N = 1024 with the Gaussian filtering method is inaccurate compared to the result with the unfiltered spectral method with N = 128 or N = 64. Figure 18 shows the waveform in logarithmic scale for t ∈ [2000, 2070]. The left figure shows the unfiltered spectral approximations with N = 128 (black), N = 64 (red), N = 48 (blue), and N = 32 (magnet). The right figure shows the filtered finite difference approximations with N = 1024 (black), N = 128 (red),N = 64 (blue) and N = 32 (green). The figures clearly show that the spectral approximation yields a fast convergent behavior as N is increased. The figures also show that the spectral approximation yields decaying waveforms 29
0
0.8
−1
0.7 0.6
Log10|ψ/m0|
Log10|ψ/m0|
−2 −3 −4 −5
0.4 0.3 0.2
−6 −7 −1.5
0.5
0.1
−1
−0.5
0
0.5
1
0 1.5
1.5
2
Log10t
2.5
Log10t −3 −4 −5
0.4 0
Log |ψ/m |
0.3
−6
10
Log10|ψ/m0|
0.5
−7
0.2 −8
0.1
0 2.5
−9
2.55
2.6
2.65
2.7
2.75
2.8
Log10t
−10 3.2
3.25
3.3
3.35
3.4
Log t 10
Fig. 17. Log10 |ψ/m0| versus Log10 t with different time intervals. Top left: t ∈ [0.0316, 31.6228]. Top right: t ∈ [31.6228, 316.2278]. Bottom left: t ∈ [316.2278, 630.9573]. Bottom right: t ∈ [1584.893, 2000]. The black, red, blue and solid lines represent the finite difference method with the Gaussian filtering with N = 1024, 128, 64, and 32, respectively.
even for large t without any filtering method applied. The finite difference approximations, however, do yield only a slow convergence as the right figure indicates. Here note that the convergence with the finite difference method is only obtained with the Gaussian filtering. The numerical results given in Figures 16 and 18 indicate that the spectral approximation yields a stable and accurate result of the waveform for long time integration.
5
Conclusion
In this work, the inhomogeneous Zerilli equation has been solved numerically both with the Chebyshev spectral collocation method and the 2nd order explicit finite difference method to investigate the waveforms for head-on collisions of black holes in one dimension. Various numerical results have been presented. The infalling black hole is considered as a point source and consequently singular source terms appear in the governing equations. For the 30
−1 N = 32
N = 32
−2 N = 64 −3
Log10|ψ/m0|
0
Log | ψ/m |
−4
−5
10
−4 N = 128
−5
−6
N = 128 N = 64 N = 48
−6 N = 1024
−7 2000
−7
2020
2040
2060
2000
t
2010
2020
2030
2040
2050
2060
2070
t
Fig. 18. Log10 |ψ/m0| versus t. t ∈ [2000, 2070]. Left: Unfiltered spectral approximations with N = 128(black), 64(red), 48(blue), and 32(magnet). Right: Filtered finite difference approximations with N = 1024 (black), 128 (red), 64 (blue), and 32 (green).
approximation of singular source terms we use the direct derivative projection of the Heaviside function on the collocation points. Such approximation yields very oscillatory approximations of the δ-function and its derivative(s). Despite these oscillations, the spectral solution is obtained in an accurate way as the direct derivative projection is consistently formulated with the given PDEs. There are many nonlinear PDEs with the singular source terms such as those related to shock phenomenon in fluid mechanical problems. The direct projection method considered in this paper has been applied only to linear problems and further extension of the method to the nonlinear case will be investigated in our future work. The simple wave equation in 1 + 1 spacetime shows that the spectral method recovers the exponential convergence even with the local jump discontinuities. The same technique is applied to solve the Zerilli equation and various numerical results have been obtained. These results are also compared with the second-order explicit finite difference method. The spectral solutions with various N clearly show the gravitational ringdown waveforms and also capture the decaying waveforms for large time t. The spectral approximation with large N shows small oscillatory behavior after the effect of the initial oscillations is minimized. To reduce these unphysical oscillations, the spectral filtering method is also applied. The spectral method with the proper order p of filtering, yields a smooth waveform with these oscillations considerably reduced. After black hole ringdown starts, all the spectral approximations agree well with each other and exhibit nice decaying profiles. Early in the evolution, the spectral filtering has an effect that the nonphysical oscillations due to the inconsistent initial condition are amplified. Such amplification effect is not explained in this paper and will be investigated in the future work. The initial oscillations leave the computational domain within a finite time. Thus, these oscillations have minimal effects on the overall waveform and the spectral methods with different N and p, yield results in good agreement. The results from the finite difference method are highly oscillatory 31
for large N. Furthermore, the spectral method yields highly accurate results for large t even though no filtering is applied. Approximations with the finite difference method differ very much from those with the spectral method especially for large time t. The finite difference method needs large N to obtain the approximation with the same accuracy of the spectral method. Even with large N the filtering method is necessary for the finite difference method while the unfiltered spectral solution still yields an accurate power-law decay. The spectral method, yields an accurate result with smaller N and can be efficiently used for the approximation of these types of problems. Our future work will center around the development of more efficient spectral methods such as the spectral method with the domain decomposition technique and adaptive filtering method for the improvement of accuracy and computational efficiency. As for the multi-domain spectral method or the discontinuous Galerkin method, there is a possibility to place the singular source term at the domain or element interface so that the singular source term plays a role only as a boundary or interface condition. In this case, the singular source term is involved in the given PDEs only in a weak sense. For the spectral multi-domain penalty method, the penalty parameter can be also exploited to enhance both stability and accuracy associated with the singular source term. In general, however, the location of the singular source term varies with time and it would be more convenient to allow the singular source term to exist inside the sub-domain or element, in particular when the range of the possible locational variation is large. For this reason, the current study with the single spectral domain will be important for our future research. We will also investigate more generalized collisional system of black holes and compare with the results from the frequency domain method. ACKNOWLEDGMENTS JHJ was supported by the NSF under Grant No. DMS-0608844. GK and IN acknowledge research support from the UMD College of Engineering RSIF and the Physics department.
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