- NUMERICAL SIMULATION OF DEGENERATE NECKPINCHES 311
Imposing a restriction like rotational symmetry on the geometries under study
is useful only if the restriction is preserved by the Ricci flow equations. As noted in
§8 of Chapter 4 in [77], Ricci flow does indeed preserve such Lie group generated
isometries. Hence we may proceed to calculate the Ricci flow equations and evaluate
their evolution for the problem at hand working solely in terms of the functions
X('ljJ, t) and W('ljJ, t), or, equivalently, X('ljJ, t) and S('ljJ, t). The Ricci flow system
becomes a 1 + 1 initial value boundary value problem for these two functions on
(0, 7r) x lR+ We note that, in reducing the system in this way, the rotational
symmetry assumption allows us to avoid confronting the problem of doing numerical
analyses on a nontrivial m anifold.
As discussed in Chapter 3 of Volume One, the Ricci flow PDE system itself
is not parabolic, but one can prove that its initial value problem is well-posed by
modifying the system via the addition of a Lie derivative term to obtain a Ricci-
DeTurck flow PDE system, which is parabolic, and then using the well-posedness
of the Ricci- DeTurck flow initial value problem together with the fact that Ricci
flow solutions and Ricci-DeTurck flow solutions are related by well-defined diffeo-
morphisms. One might guess that while the Ricci- DeTurck flow is useful as a tool
for proving the well-posedness of the Ricci flow, in numerical simulations one can
work directly with the Ricci flow itself. Experience shows that this is not the case
and that the Ricci- DeTurck flow is just as important for numerical simulations. In
particular, one finds in carrying out the numerical simulation for the (rotationally
symmetric) critical neckpinch study that the strictly parabolic Ricci- DeTurck flow
system is numerically stable, while the weakly parabolic Ricci flow system is not.
In retrospect, this makes sense since strict parabolicity is useful in proving well-
posedness because it allows one to use energy estimates to prove convergence of
successive approximations, and of course this same sort of convergence is crucial to
the success of a numerical simulation.
Since the Ricci- DeTurck flow is needed for the critical neckpinch study, we
now display the evolution equations for X('ljJ, t) and S('ljJ, t) corresponding to such
a flow^2 , also including a volume normalizing term:
ax = e^2 (W-X) [x" + 2cot'ljJX' - 2 + ~ ((X')^2 + (W')^2 ) + 3X'W'
at 2(36.2) + (1 - e-^4 w) (~ + 1+2cot'ljJW') J + _
3
f ,
2sm 'ljJaS = e^2 (W-X) [s" + 6cot'ljJS' - 8S - -
3
- (1-4W - e-^4 W)
at 2sin^4 '1j;
1 -4W
+ -. e 2 (1 - 2 (cot 'ljJX' + 2 sin 'ljJ cos 'ljJS' + 4cos^2 '1/JS))
sm 'ljJ(36.3) - ~ ( ( ~' )2
+(sin 'ljJS' + 2 cos 'ljJS)^2 + ~X.',. (sin 'ljJS' + 2 cos 'ljJS))].
2 sm'lj; sm '!-'2 As described in Chapter 3 of Volume One, there is not a unique Ricci-DeTurck flow; for each
choice of a background connection, there is a different version of it. All of these are equally useful
for proving well-posedness of the Ricci flow. It may, however, turn out that different versions of
Ricci-DeTurck flow have different numerica l stability properties, and therefore different versions
may be useful for different numerica l simulations.