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.