Coordinates

Illinois codes are designed to be coordinate-agnostic. In practice, this means knowledge of the geometry is abstracted to two functions: a "coord" function which maps from logical (internal grid) coordinates to Kerr-Schild ones and also a "get covariant metric" function which populates double gcov[4][4], the covariant metric, as a function of position on the logical grid. Auxiliary functions then provide other functionality (like the connection coefficients by performing numerical derivatives).

For our black hole simulations, we use one of several modified versions of Kerr-Schild coordinates for our logical coordinates. They are explained briefly below:

Kerr-Schild "ks"

These are the standard coordinates x^a = (t,r,θ,Φ) used to express the Kerr metric. Compared to Boyer-Lindquist coordinates, ks coordinates are regular on the horizon. In ks coordinates, the components of the covariant metric g_ab are

g_00 = -1 + 2 r / Σ
g_11 = 1 + 2 r / Σ
g_22 = Σ
g_33 = sin^2(θ) ( Σ + a^2 sin^2(θ) (1 + 2 r / Σ) ) 
g_01 = g_10 = 2 r / Σ
g_03 = g_30 -2 a r sin^2(θ) / Σ
g_13 = g_31 = - a sin^2(θ) (1 + 2 r / Σ)

Σ = r^2 + a^2 cos^2(θ)

with all other components zero.

The metric g_μν for any of the following modified versions of ks coordinates can be computed by performing a coordinate transformation (using the provided relations) as

g_μν = Λ^a_μ Λ^b_ν g_ab

where g_ab is the metric given above in ks coordinates and the components of the transformation matrices Λ are defined by

Λ^a_μ = ∂ x^a / ∂ x^μ

Modified Kerr-Schild "mks"

Metric parameters for mks coordinates are a and hslope. Logical mks coordinates x^μ = (x⁰,x¹,x²,x³) are related to ks coordinates by

t = x0
r = exp(x1)
θ = π x2 + (1 - hslope)/2 sin(2 π x2)
Φ = x3

By increasing the value of hslope, grid zones can be concentrated about the midplane where much of the disk physics happens, thereby increasing effective physical resolution without increasing the logical number of zones.

Funky Modified Kerr-Schild "fmks"

In addition to the a and hslope of mks coordinates, metric parameters for fmks include mks_smooth, poly_xt, and poly_alpha here written as A, B, and C respectively. Logical fmks coordinates x^μ = (x⁰,x¹,x²,x³) are related to ks coordinates by

t = x0
r = exp(x1)
θ = θg + exp(- A Dx1) (θj - θg)
Φ = x3

where

θg = π x2 + (1 - hslope)/2 sin(2 π x2)
θj = D (2 x2 - 1) (1 + ((2 x2 - 1)/B)^C/(1 + C) ) + π / 2
D = π / ( 2 + 2 / ( B^C + C B^C )  )

and where Dx1 = x1 - startx1 is the displacement (in mmks) from the left edge of the logical grid.

fmks coordinates decrease grid density in polar regions to effectively decrease the amount of time required to evolve the non-critical physics there.