312 36. TYPE II SINGULARITIES AND DEGENERATE NECKPINCHES
Here we note that the average scalar curvature f is given by
(36.4)
f = ~ la"' d'lj;ex+^3 w (e-^4 w -1-4sin'lj;cos'¢W' + sin^2 '¢[3 + (X' + W')
2
J),
where the normalization constant N is given by(36.5) N =la"' d'lj; e^3 x+w sin^2 '¢.
With the evolution equations in hand, the next task is to choose a I-parameter
family of initial data geometries which correspond to the variable neckpinches of
interest. To do this, one may set W = X, and one may then choose X so that
4e^4 x sin^2 '¢ = sin^2 2'¢ for cos^2 '¢ ~ 1 /2 and 4e^4 x sin^2 '¢ = sin^2 2'¢ + 4,\cos^2 2'¢ for
cos^2 '¢ ::; 1 /2. Here ,\, which is a constant taking values between 0 and +oo, serves
as the parameter, and we note that for ,\ = 0 this data represents two round 3-
spheres joined at the poles, while for ,\ sufficiently large, the central cusp smooths
out and the geometry more closely approximates that of a single round sphere.
1.2.2. Numerical considerations.
At least three main approaches are commonly used for carrying out the explicit
numerical simulation of solutions of partial differential equation systems on mani-
folds. Most traditional is partial differencing; also widely used are spectral methods
and finite-element techniques. Each of these approaches has advantages and dis-
advantages, each has strong advocates, and all three have been used in numerical
relativity, with partial differencing being the predominant choice in that field.
For the case being described here- rotationally symmetric solutions of the Ricci
fl.ow equations on S^3 - the spatial domain reduces to an interval of the real line, and
consequently partial differencing is very simple to implement, is not very expensive
in computer time, and is expected to be sufficiently accurate. Hence that is the
approach which has been used. We now outline how it works for this application.
One starts by dividing the spatial coordinate range (0, 7r) of'¢ into N - 2 pieces,
so that one has 6.'¢ = n/(N - 2). One then chooses N grid points, including a
pair which run outside the coordinate range. Hence the first spatial grid point is at
'¢ = -6.'¢/2, while the last is at'¢= 7r + 6.'¢/2. Now in terms of these grid points,
a function of the form F('lj;, t 0 ) for fixed time t 0 is replaced by a set of N numbers
Fi = F ((i - ~)/::,.'¢,to) where 1 ::; i ::; N, and spatial derivatives are replaced by
centered finite differences in the following way:
(36.6) 8F Fi+1 - Fi-1
8'¢ ---+ 26. '¢ ,
82 F Fi+1 + Fi-1 - 2Fi
-----+ ----~--
8¢2 (6.'¢)2(36. 7)
For the time dependence of these functions, one chooses a fixed time step 6.t and
replaces F (( i - ~) /::,. '¢, n6.t) by the numbers Ft.
Now, for an evolution equation of the form ~~ ('¢, t) = G('lj;, t) one numerically
evolves using the approximation
(36.8)This evolution is implemented for all values of i except 1 and N. Note that these
two "ghost zones" are not part of the manifold since '¢ is not in the range 0 ::; '¢ ::; n.