Toward computing gravitational initial data without elliptic solvers

Abstract

Two new methods have been proposed for solving the gravitational constraints without using elliptic solvers by formulating them as either an algebraichyperbolic or parabolic-hyperbolic system. Here, we compare these two methods and present a unified computational infrastructure for their implementation as numerical evolution codes. An important potential application of these methods is the prescription of initial data for the simulation of black holes. This paper is meant to support progress and activity in that direction.

Numerical construction of initial data sets of binary black hole type using a parabolic-hyperbolic formulation of the vacuum constraint equations

“…We have not compared this to any other method (for example, the Newton method) yet. 3.4 The parabolic hyperbolic equations written in terms of the SWSH can be found in [37] is of (pseudo-)spectral nature. All examples considered so far assume axial symmetry, so there is no dependence on ϕ.…”

Numerical investigations of the asymptotics of solutions to the evolutionary form of the constraints

“…Nevertheless, in either case we can represent all the tangential derivatives via the operators ð and ð (if needed supplemented by the use of ∂ φ ) and a very convenient set of variables can also be introduced. This latter is done by fixing a complex dyad {q i , q i } on the unit sphere S 2 which, as proposed in [12,13,14], can be mapped first onto one of the leaves (say onto S 0 ) and in the second step Lie propagate onto all the S r leaves along the flow r i . Having the complex dyad {q i , q i } defined throughout Σ we will use instead of the variables ( N , N A , γ AB , κ, k A , K E E ,…”

“…These new variables possess definite spin-weights whence they are also analogous to the basic variables applied in the Newman-Penrose formalism. They were defined in [12,13,14] but for convenience of the readers they are also recalled in Table 1.…”

“…The variables used in recasting the constraints in [12,13,14]. For detailed derivations for some of these and some other more involved expressions see references [12,13,14].…”

Asymptotically flat vacuum initial data sets from a modified parabolic-hyperbolic formulation of the Einstein vacuum constraint equations

“…The system Eqs. (3.1)-(3.3) has been used in several works among which are [12,19,22,28,35]. The particular choice of how to split the fields into free data and unknowns is however not the only possibility.…”