We show that by adding suitable lower-order terms to the Z4 formulation of the Einstein equations, all constraint violations except constant modes are damped. This makes the Z4 formulation a particularly simple example of a λ-system as suggested by Brodbeck et al. We also show that the Einstein equations in harmonic coordinates can be obtained from the Z4 formulation by a change of variables that leaves the implied constraint evolution system unchanged. Therefore the same method can be used to damp all constraints in the Einstein equations in harmonic gauge.
In the Cauchy problem of general relativity one considers initial data that satisfies certain constraints. The evolution equations guarantee that the evolved variables will satisfy the constraints at later instants of time. This is only true within the domain of dependence of the initial data. If one wishes to consider situations where the evolutions are studied for longer intervals than the size of the domain of dependence, as is usually the case in three dimensional numerical relativity, one needs to give boundary data. The boundary data should be specified in such a way that the constraints are satisfied everywhere, at all times. In this paper we address this problem for the case of general relativity linearized around Minkowski space using the generalized Einstein-Christoffel symmetric hyperbolic system of evolution equations. We study the evolution equations for the constraints, specify boundary conditions for them that make them well posed and further choose these boundary conditions in such a way that the evolution equations for the metric variables are also well posed. We also consider the case of a manifold with a non-smooth boundary, as is the usual case of the cubic boxes commonly used in numerical relativity. The techniques discussed should be applicable to more general cases, as linearizations around more complicated backgrounds, and may be used to establish well posedness in the full non-linear case.
This is the first paper in a series aimed to implement boundary conditions consistent with the constraints' propagation in 3D unconstrained numerical relativity. Here we consider spherically symmetric black hole spacetimes in vacuum or with a minimally coupled scalar field, within the Einstein-Christoffel (EC) symmetric hyperbolic formulation of Einstein's equations. By exploiting the characteristic propagation of the main variables and constraints, we are able to single out the only free modes at the outer boundary for these problems. In the vacuum case a single free mode exists which corresponds to a gauge freedom, while in the matter case an extra mode exists which is associated with the scalar field. We make use of the fact that the EC formulation has no superluminal characteristic speeds to excise the singularity.We present a second-order, finite difference discretization to treat these scenarios, where we implement these constraint-preserving boundary conditions, and are able to evolve the system for essentially unlimited times. As a test of the robustness of our approach, we allow large pulses of gauge and scalar field to enter the domain through the outer boundary. We reproduce expected results, such as trivial (in the physical sense) evolution in the vacuum case (even in gauge-dynamical simulations), and the tail decay for the scalar field.
We extend the notion of numerical stability of finite difference approximations to include hyperbolic systems that are first order in time and second order in space, such as those that appear in numerical relativity and, more generally, in Hamiltonian formulations of field theories. By analyzing the symbol of the second order system, we obtain necessary and sufficient conditions for stability in a discrete norm containing one-sided difference operators. We prove stability for certain toy models and the linearized Nagy-Ortiz-Reula formulation of Einstein's equations.We also find that, unlike in the fully first order case, standard discretizations of some well-posed problems lead to unstable schemes and that the Courant limits are not always simply related to the characteristic speeds of the continuum problem. Finally, we propose methods for testing stability for second order in space hyperbolic systems.
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