A.9 Appendix B: Numerical methods for evolutions
A.9.2 BSSN-moving-puncture evolutions
the inertial coordinatesxi. This map is given by
x = rsinθ00cosφ00, (A.44)
y = rsinθ00sinφ00+e−r002/r002T Y(t), (A.45)
z = rcosθ00, (A.46)
r = r˜ [
1 + sin2(π˜r/2R00max)
× (
A(t)Rmax0
Rmax00 + (1−A(t)) R0max3 R00maxR002 −1
)]
, (A.47)
˜
r = r00−q(r00)
`∑max
`=0
∑` m=−`
λ`m(t)Y`m(θ00, φ00), (A.48)
(r00, θ00, φ00) are spherical polar coordinates in the new comoving coordinate system,R00maxis the value ofr00 at the outer boundary, andrT00 is a constant chosen to be 31.21MADM. The function q(r00) is given by
q(r00) =e−(r00−R00AH)3/σ3q, (A.49) where R00AH is the desired radius of the common apparent horizon in comoving coordinates. The functionA(t) is
A(t) =A0+ (A1+A2(t−tm))e−(t−tm)/τA, (A.50) where the constants A0, A1, and A2 are chosen so that A(t) matches smoothly onto a(t) from Eq. (A.36): A(tm) = a(tm), ˙A(tm) = ˙a(tm), and ¨A(tm) = ¨a(tm). The constantτA is chosen to be on the order of 5M. The functions Y(t) and λ`m(t) are determined by dynamical control systems that keep the apparent horizon spherical and centered at the origin in comoving coordinates; see [4]
for details.
confirm their Eq. (26) as the most efficient method to evolve the shift vector. In contrast to the shift, moving puncture codes show little variation in the evolution of the lapse function. Here we follow the most common choice so that our gauge conditions are given by
∂tα = βi∂iα−2αK, (A.51)
∂tβi = βm∂mβi+3 4
Γ˜i−ηβi. (A.52)
Γ˜i is the contracted Christoffel symbol of the conformal 3-metric, K the trace of the extrinsic curvature [see for example Eq. (1) of [72]] andη a free parameter set to 1 unless specified otherwise.
For further details about the moving puncture method and the specific implementation in theLean code code we refer to Sec. II of Ref. [72]. Except for the use of sixth instead of fourth order spatial discretization [101], we did not find it necessary to apply any modifications relative to the simulations presented in that work.
The calculation of the 4-momentum in the Lean code is performed in accordance with the relations listed in Sec. A.3. The only difference is that in a BSSN code the four metric and its derivatives are not directly available but need to be expressed in terms of the 3-metric γij, the extrinsic curvatureKij as well as the gauge variables lapseαand shiftβi. The key quantity for the calculation of the 4-momentum is the integrand in Eq. (A.7). A straightforward calculation gives it in terms of the canonical ADM variables
∂αH0α0j = 1 χ3
[3
χγjm∂mχ+γkmγjn∂kγmn ]
, (A.53)
∂αHiα0j = 1 χ3
[2α(Kij−γijK) +γij∂mβm−γimγmβj]
−βi∂αH0α0j, (A.54)
where K:=Kii andχ:= detγ−1/3 have been used for convenience because they are fundamental variables in our BSSN implementation.
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