So far we were concerned with describing mathematically well-defined procedures
for constructing initial data (gij, Kij). Both the conformal thin sandwich equations
data set, and in chapter 3 of this thesis we will present a numerical code to solve these equations.
Having these mathematical and numerical tools, we now face the physical ques- tion how to choose the free data such that the final physical initial data sets have certain properties. Moreover, the presence of black holes often leads to excised regions from the computational domain, which require boundary conditions. For elliptic equations, boundary conditions influence the solution everywhere, so one must choose them with great care.
Both questions –how to choose the free data and how to choose boundary
conditions at the excised regions5– are difficult, because we do not know exactly
how these choices influence the physical solution, neither do we know what the physical solution “should” be for certain astrophysically relevant situations, for example an inspiraling binary black hole.
The approach taken in this thesis is to compute initial data sets with both formalisms and with several different choices of free data. Then we look for con- sistency among these initial data sets. Particularly helpful are sequences of initial data sets. For example, we compute initial data representing binary black holes in quasi-circular orbits as a function of the separation between the black holes. Anal- ysis of such a sequence, and verification that certain properties along the sequence seem physically “reasonable,” can increase of decrease the confidence one has in the relevance of the particular approach used to construct the sequences.
In the rest of this section, we briefly review the choices made in each of the subsequent chapters to put them into context. From now on, we focus exclusively
on vacuum spacetimes with vanishing matter terms, ρ = ji = S
ij = 0, containing
one or two black holes.
The project presented in chapter 4 was begun before the modern constraint decompositions were discovered. Therefore it uses one of the older methods, the “Conformal transverse traceless decomposition” [4, 20]. It is a special case of the extrinsic curvature decomposition outlined in section 2.3 which is obtained
by setting ˜σ ≡ 1. It further assumes the simplest possible free data: conformal
flatness, ˜gij = flat, maximal slicing, K = 0, and ˜AijT T= 0. The momentum constraint
(2.54) decouples from the Hamiltonian constraint (2.55), and simplifies to ˜
∇j(˜LV )ij = 0, (2.71)
where in Cartesian coordinates, the derivatives are simple partial derivatives.
Bowen and York [21, 22] found analytic solutions, Vi
BY, to (2.71) describing one or
multiple black holes carrying linear and angular momenta.
The initial value problem is thus reduced to solving the Hamiltonian constraint (2.55), which becomes
∆ψ + 1
8ψ
−7(˜LV
BY)ij(˜LVBY)ij = 0, (2.72)
a quasi-linear flat space Laplace equation.
The boundary conditions at the black holes are derived by requiring inversion symmetry with respect to the throat of each black hole. This leads to a two-sheeted topology, where each black hole connects “our” universe through an Einstein- Rosen bridge to the same “mirror” universe [9]. To satisfy inversion symmetry,
˜
AijBY = (˜LVBY)ij must be modified by an ingenious imaging process [23]6. This
method is described in detail by Cook [26, 27, 28]. Cook also developed an elliptic solver for this problem [28], which I use in chapter 4.
Chapter 5 is a variation on chapter 4, exploring quasi-circular orbits for unequal mass black holes (as opposed to spinning black holes). It uses the inversion sym- metric Bowen-York data to construct sequences of quasi-circular orbits extending toward the test-mass limit. This limit is known analytically, so one can compare directly the computations against the correct results.
The “Bowen-York initial data” is relatively easy to construct (one flat space Laplace equation instead of four or five coupled equations; moreover, the puncture method [24] does not even require internal boundaries). However, it makes very
special assumptions about the free data, ˜gij = flat, K = 0, ˜AijT T = 0, ˜σ = 1.
Only a small subset of all initial data sets can be reached with these restrictive assumptions. As it is not clear that realistic binary black hole data belongs into this class, one needs to go beyond this approach.
In chapter 6, we compare different constraint decompositions, namely the two previous variants of the extrinsic curvature decomposition, as well as the conformal
thin sandwich equations with free data (2.30) (specification of ˜N , not ˙K). We also
6The puncture method [24] makes the same assumptions on the free data, but
uses a three-sheeted topology instead, with each black hole connecting to “its own” universe (cf. [10]). This approach does not require excised regions in the compu- tational domain. A very recent paper [25] indicates that the puncture method cannot be easily extended to the conformal thin sandwich equations.
explore different choices for some of the free data by choosing ˜gij and K (and ˜N
for conformal thin sandwich) based on superposed Kerr-Schild data.
Chapter 7 employs the conformal thin sandwich formalism with specification of ˙
K, i.e. we solve the five coupled partial differential equations (2.28), (2.29), (2.35). We also explore different boundary conditions at the horizons of the black holes.
Chapter 8, finally, explores spacetimes without black holes, or with just one.
We use the conformal thin sandwich equations with ˙K-equation to construct initial
data for a spacetime with superposed gravitational radiation. In this chapter, we solve the conformal thin sandwich equations on very general, nonflat conformal
A multidomain spectral method for
solving elliptic equations
∗
3.1
Introduction
Elliptic partial differential equations (PDE) are a basic and important aspect of almost all areas of natural science. Numerical solutions of PDEs in three or more dimensions pose a formidable problem, requiring a significant amount of memory and CPU time. Off-the-shelf solvers are available; however, it can be difficult to adapt such a solver to a particular problem at hand, especially when the compu- tational domain of the PDE is nontrivial, or when one deals with a set of coupled PDEs.
There are three basic approaches to solving PDEs: Finite differences, finite elements and spectral methods. Finite differences are easiest to code. However, they converge only algebraically and therefore need a large number of grid points and have correspondingly large memory requirements. Finite elements and spec- tral methods both expand the solution in basis functions. Finite elements use many subdomains and expand to low order in each subdomain, whereas spectral methods use comparatively few subdomains with high expansion orders. Finite elements are particularly well suited to irregular geometries appearing in many en- gineering applications. For sufficiently regular domains, however, spectral methods are generally faster and/or more accurate.
Multidomain spectral methods date back at least to the work of Orszag[29]. In a multidomain spectral method, one has to match the solution across different subdomains. Often this is accomplished by a combination of solves on individual subdomains together with a global scheme to find the function values at the in- ternal subdomain boundaries. Examples of such global schemes are relaxational iteration[30], an influence matrix[31, 32], or the spectral projection decomposition method[33]. For simple PDEs like the Helmholtz equation, fast solvers for the sub- domain solves are available. For more complicated PDEs, or for coupled PDEs, the subdomain solves will typically use an iterative solver. One drawback of these
∗H. P. Pfeiffer, L. E. Kidder, M. A. Scheel and S. A. Teukolsky, Comput. Phys.
Commun. 152, 253 (2003).
schemes is that information from the iterative subdomain solves is not used in the global matching procedure until the subdomain solves have completely converged. The question arises whether efficiency can be improved by avoiding this delay in communication with the matching procedure.
In this paper we present a spectral method for coupled nonlinear PDEs based on pseudospectral collocation with domain decomposition. This method does not split subdomain solves and matching into two distinct elements. Instead it com- bines satisfying the PDE on each subdomain, matching between subdomains, and satisfying the boundary conditions into one set of equations. This system of equa- tions is then solved with an iterative solver, typically GMRES[34]. At each itera- tion, this solver thus has up-to-date information about residuals on the individual subdomains and about matching and thus can make optimal use of all information. The individual subdomains implemented are rectangular blocks and spheri- cal shells. Whereas either rectangular blocks (see e.g. [35, 36, 37]) or spherical shells[38] have been employed before, we are not aware of work using both. The code supports an arbitrary number of blocks and shells that can touch each other and/or overlap.
Moreover, the operatorS at the core of the method (see section 3.3.3) turns out
to be modular, i.e. the code fragments used to evaluate the PDE, the boundary conditions, and the matching conditions are independent of each other. Thus the structure of the resulting code allows for great flexibility, which is further enhanced by a novel point of view of the mappings that are used to map collocation coor- dinates to the physical coordinates. This flexibility is highlighted in the following aspects:
• The user code for the particular PDE at hand is completely independent from the code dealing with the spectral method and domain decomposition. For a new PDE, the user has to supply only the code that computes the residual and its linearization.
• Mappings are employed to control how collocation points and thus resolution are distributed within each subdomain. New mappings can be easily added which are then available for all PDEs that have been coded.
• The solver uses standard software packages for the Newton-Raphson step, the iterative linear solvers, and the preconditioning. Thus one can experi-
ment with many different linear solvers and different preconditioners to find an efficient combination. The code will also automatically benefit from im- provements to these software packages.
• The code is dimension independent (up to three dimensions).
• Many properties of a particular solve can be chosen at runtime, for example the domain decomposition, the mappings used in each subdomain, as well as the choice of the iterative solver. The user can also choose among differential operators and boundary conditions previously coded at runtime.
In the next section we recall some basics of the pseudo-spectral collocation method. In section 3.3 we describe our approach of combining matching with solving of the PDE. For ease of exposition, we interweave the new method with more practical issues like mappings and code modularity. The central piece of
our method, the operator S, is introduced in section 3.3.3 for a one-dimensional
problem and then extended to higher dimensions and spherical shells. In section 3.4 we solve three example problems. Many practical issues like preconditioning and parallelization are discussed in this section, and we also include a detailed comparison to a finite difference code.