Polarized plane symmetric cosmologies have been evolved numerically using standard techniques by Anninos, Centrella, and Matzner [2, 3]. A long-term project involving Berger, Garfinkle, and Moncrief and our students to study the generic cosmological singularity numerically has been reviewed in detail elsewhere  but will be discussed briefly here.
Einstein's equations split into two groups. The first is nonlinearly coupled wave equations for dynamical variables P and Q (where ) obtained from the variation of 
This Hamiltonian has the form required by the symplectic scheme. If the model is, in fact, AVTD, the approximation in the symplectic numerical scheme should become more accurate as the singularity is approached. The second group of Einstein equations contains the Hamiltonian and -momentum constraints respectively. These can be expressed as first order equations for in terms of P and Q . This break into dynamical and constraint equations removes two of the most problematical areas of numerical relativity from this model--the initial value problem and numerical preservation of the constraints.
For the special case of the polarized Gowdy model (Q =0), P satisfies a linear wave equation whose exact solution is well-known . For this case, it has been proven that the singularity is AVTD . This has also been conjectured to be true for generic Gowdy models .
AVTD behavior is defined in  as follows: Solve the Gowdy wave equations neglecting all terms containing spatial derivatives. This yields the AVTD solution . If the approach to the singularity is AVTD, the full solution comes arbitrarily close to an AVTD solution at each spatial point as . As , the AVTD solution becomes
where v > 0. Substitution in the wave equations shows that this behavior is consistent with asymptotic exponential decay of all terms containing spatial derivatives only if . We have shown that, except at isolated spatial points, the nonlinear terms in the wave equation for P drive v into this range [25, 26]. The exceptional points occur when coefficients of the nonlinear terms vanish and are responsible for the growth of spiky features seen in the wave forms [32, 25]. We conclude that generic Gowdy cosmologies have an AVTD singularity except at isolated spatial points [25, 26]. A claim to the contrary  is incorrect . Recently, it has been proven analytically that Gowdy solutions with 0 < v < 1 and AVTD behavior almost everywhere are generic .
Addition of a magnetic field to the vacuum Gowdy models (plus a topology change) which yields the inhomogeneous generalization of magnetic Bianchi VI models provides an additional potential which grows exponentially if 0 < v < 1. Local Mixmaster behavior has recently been observed in these magnetic Gowdy models .
Here and are analogous to P and Q while is a conformal factor for the metric in the u - v plane perpendicular to the symmetry direction. Note particularly that contains two copies of the Gowdy plus a free particle term and is thus exactly solvable. The potential term is very complicated. However, it still contains no momenta, so its equations are trivially exactly solvable. However, issues of spatial differencing are problematic. (Currently, a scheme due to Norton  is used. A spectral evaluation of derivatives  which has been shown to work in Gowdy simulations  does not appear to be helpful in the U (1) case.) A particular solution to the initial value problem is used, since the general solution is not available . Currently, except as discussed below, the constraints are monitored but not explicitly enforced.
Despite the current limitations of the U (1) code, conclusions can be confidently drawn for polarized U (1) models in our restricted class of initial data. No numerical difficulties arise in this case. Polarized models have . Grubisic and Moncrief  have conjectured that these polarized models are AVTD. The numerical simulations provide strong support for this conjecture [26, 31].
Mixmaster simulations with the new algorithm  can easily evolve more than 250 bounces reaching . This compares to earlier simulations yielding 30 or so bounces with . The larger number of bounces quickly reveals that it is necessary to enforce the Hamiltonian constraint. A similar conclusion for gravitational wave equations was obtained by Gundlach and Pullin . Although this particular analysis may be incorrect , it is still likely that constraint enforcement will be essential for sufficiently long simulations. An explicitly constraint enforcing U (1) code was developed some years ago by Ove (see  and references therein).
|Numerical Approaches to Spacetime Singularities
Beverly K. Berger
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