Solving the Einstein constraints in periodic spaces with a multigrid approach

Abstract

Abstract. Novel applications of Numerical Relativity demand for more flexible algorithms and tools. In this paper, I develop and test a multigrid solver, based on the infrastructure provided by the Einstein Toolkit, for elliptic partial differential equations on spaces with periodic boundary conditions. This type of boundary often characterizes the numerical representation of cosmological models, where space is assumed to be made up of identical copies of a single fiducial domain, so that only a finite volume (with periodic boundary conditions at its edges) needs to be simulated. After a few tests and comparisons with existing codes, I use the solver to generate initial data for an infinite, periodic, cubic black-hole lattice.

A Gravitational Wave Background from Primordial Black Hole Lattices in Matter Dominated Era*

“…And there are no outer boundary conditions for PBHLs. Although the set up of initial data is usually a non-trivial task, here we can solve PBHLs' coupled system of one non-linear and three linear elliptic partial differential equations thanks to [10,12,15].…”

“…We use a multigrid approach [15] to solving the Einstein constraints (6). But we impose the integrability condition as [10] 2πm PBH − 1 12…”

Black hole universe with a cosmological constant

“…The black hole lattice universe is often used as one of tools to evaluate effects of local non-linear inhomogeneities on the global expansion. Recently, black hole lattice universe models have been revisited by several authors [2][3][4][5][6][7][8][9][10][11][12]. Time symmetric initial data for N-black hole systems on a virtual 3-sphere have been analyzed in Refs.…”

“…Initial data for a black hole inside a cubic box with a periodic boundary condition have been constructed and analyzed in Refs. [3,8], and those time evolutions have been investigated in Refs. [2,7].…”

Constant-mean-curvature slicing of the Swiss-cheese universe

“…Numerical simulations of spacetime dynamics in cosmological settings have been actively performed in recent years. One main motivation to simulate the cosmological nonlinear dynamics is to quantify the effect of the non-linear small scale inhomogeneity on the global expansion law of the universe [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Another significant motivation comes from primordial black holes [17,18].…”