Numerical Relativity Using a Generalized Harmonic Decomposition

TL;DR

A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced and a variant of the cartoon method for efficiently simulating axisymmetric spacetimes with a Cartesian code is described.

Abstract

A new numerical scheme to solve the Einstein field equations based upon the generalized harmonic decomposition of the Ricci tensor is introduced. The source functions driving the wave equations that define generalized harmonic coordinates are treated as independent functions, and encode the coordinate freedom of solutions. Techniques are discussed to impose particular gauge conditions through a specification of the source functions. A 3D, free evolution, finite difference code implementing this system of equations with a scalar field matter source is described. The second-order-in-space-and-time partial differential equations are discretized directly without the use first order auxiliary terms, limiting the number of independent functions to fifteen--ten metric quantities, four source functions and the scalar field. This also limits the number of constraint equations, which can only be enforced to within truncation error in a numerical free evolution, to four. The coordinate system is compactified to spatial infinity in order to impose physically motivated, constraint-preserving outer boundary conditions. A variant of the Cartoon method for efficiently simulating axisymmetric spacetimes with a Cartesian code is described that does not use interpolation, and is easier to incorporate into existing adaptive mesh refinement packages. Preliminary test simulations of vacuum black hole evolution and black hole formation via scalar field collapse are described, suggesting that this method may be useful for studying many spacetimes of interest.

Figures (6)

• Figure 1: The discretization of a variable $f(t,x,y,z)$ in the $x-t$ plane.
• Figure 2: A depiction of the adaptive mesh structure for the convergence test simulation described in Sec. \ref{['sec_cvf']}. The image to the left corresponds to the mesh structure at $t=0$, while that to the right at $t=0.5$. The largest box in each figure, whose faces are at $i^0$, actually represent two levels of (2:1) refinement. The increase in size of the finer levels and loss of the finest level of refinement by $t=0.5$ is due to the outward propagation of the initial distributions of energy.
• Figure 3: Convergence factors (\ref{['Qdef']}) for representative grid functions from the simulation described in Sec. \ref{['sec_cvf']}. The points denote the times when $Q$ was calculated, and correspond to times when the entire grid hierarchy was in sync. Note that we only show $Q_{\bar{g}_{xx}}$ at $t=0$, as all the other functions are exactly known then, and hence $Q$ is ill-defined. This plot shows that the solution is close to second order convergent, with some caveats discussed in the text.
• Figure 4: Convergence factor of the independent residual (\ref{['ires']},\ref{['Qrdef']}) of the Einstein equations from the simulation described in Sec. \ref{['sec_cvf']}. The points denote the times when $Q_\mathcal{R}$ was calculated, and corresponds to times when the entire grid hierarchy is in sync after an evolution time step (hence there are no points at $t=0$). This plot shows we are tending toward a solution that is second order convergent.
• Figure 5: Normalized mass (\ref{['ah_mass']}) for the evolution of a vacuum $M_0=0.05$ Schwarzschild black hole in Painlevé-Gullstrand coordinates. The curve labeled $4h$ corresponds to the lowest resolution axisymmetrix simulation, while the $2h$ ($h$) curves are from axisymmetric simulations with twice (four times) the resolution. The curves $4h, 3D$ and $2h, 3D$ are from runs with identical resolution to the $4h$ and $2h$ axisymmetric simulations respectively, though the simulations were in full 3D. Note that the $2h, 3D$ simulation curve only extends till roughly $t/M_0=55$ (as we ran out of computer time then), and is effectively hidden behind the other curves as $M/M_0\approx 1$ up till then.