Consistent discretizations: The Gowdy spacetimes

We apply the consistent discretization scheme to general relativity particularized to the Gowdy spacetimes. This is the first time the framework has been applied in detail in a nonlinear generally covariant gravitational situation with local degrees of freedom. We show that the scheme can be correctly used to numerically evolve the spacetimes. We show that the resulting numerical schemes are convergent and preserve approximately the constraints as expected. Given the numerical complexity of the method, further work will be needed to make it competitive with the usual free evolution methods currently in use in numerical relativity.

Figure 1

The logic of the evolution scheme.

Figure 2

The variable $\lambda$ as a function of space ($\theta$) and time, measured by the variable $\tau$. We show two resolutions, one with 8 and another with 40 spatial points. The latter represents what is practical to run on a workstation today. One sees that the two resolutions track each other well for a while, but things deteriorate as time increases (see for instance the third line of crosses from the right). The run with 8 spatial points just does not have enough resolution to track the features of the solution in question, as can be seen from the figure.

Figure 3

The variable $P^{\tau }$ in the initial data converges to a continuum solution of the pseudoconstraints, and would converge to a solution of the continuum usual constraints of general relativity if the choice of initial data for the lapse and shift were zero.

Figure 4

The L2 norm of the shift, which shows we are approximately in a gauge with zero shift.

Figure 5

Convergence of the Riemann tensor for three different resolutions. Since we are working with an almost vanishing shift, the components of the Riemann tensor can be treated as observables and compared among different resolutions directly. With a nonvanishing dynamically determined shift this would not be possible. In such cases one can only compare observables of the theory.

Figure 6

Relative error of the Riemann tensor compared to the exact solution.

Figure 7

L2 norm of the constraints of the continuum theory evaluated in the discrete theory. As can be seen they are preserved well and in a convergent fashion. As argued in the text, the value of the constraints is a measure of the error of the evolutions in the discrete theory and this plot confirms it.

Figure 8

L2 norm of the constraints of the continuum theory evaluated in the discrete theory normalized by the factor $\exp (4\tau )$ that appears in the Hamiltonian constraint. One sees convergence with absence of growth in the normalized constraints. The reader may be puzzled as to why the curves do not start at the same point. The reason is that we are displaying the curves starting at the first iteration, not at the initial data. For very low resolutions the variables change quite a bit in the first iteration.