Algebraic stability analysis of constraint propagation

Abstract

ABSTRACT. The divergence of the constraint quantities is a major problem in computational gravity today. Apparently, there are two sources for constraint violations. The use of boundary conditions which are not compatible with the constraint equations inadvertently leads to 'constraint violating modes' propagating into the computational domain from the boundary. The other source for constraint violation is intrinsic. It is already present in the initial value problem, i.e. even when no boundary conditions have to be specified. Its origin is due to the instability of the constraint surface in the phase space of initial conditions for the time evolution equations. In this paper, we present a technique to study in detail how this instability depends on gauge parameters. We demonstrate this for the influence of the choice of the time foliation in context of the Weyl system. This system is the essential hyperbolic part in various formulations of the Einstein equations.

Towards absorbing outer boundaries in general relativity

“…For example, using maximal slicing or a slicing which insures that the mean curvature rapidly decays to zero as one approaches the outer boundary might justify criterion (iii), while forcing the normal component (with respect to the outer boundary) of the shift vector to be zero at the outer boundary guarantees (iv). On the other hand, these criteria are not justified if hyperboloidal slices are used [37,[67][68][69], where the mean curvature asymptotically approaches a constant, nonzero value. It should not be difficult to generalize our analysis to more general foliations of Minkowski spacetime using the 2 + 1 split discussed in section 2.2.…”

Implementation of standard testbeds for numerical relativity

“…Thus numerical error can be expected to lead to exponential growth of the constraint for a hyperboloidal foliation with K > 0. The situation is more complicated in the nonlinear gravitational case but similar instabilities of the system of equations governing the constraints are associated with the extrinsic curvature [30]. A negative value of K (the expanding case) tends to damp constraint violation whereas a positive value (the collapsing case) can trigger constraint violating instabilities.…”

Hyperboloidal data and evolution

“…Inspection of the equations suggests the instability to be due to the type of effect described in this section. Recently, the mathematical tools to clarify this have been discussed by Frauendiener and Vogel [16]. As a simple exercise for this type of problem, we have analyzed the case of a Maxwell field (E a , B a ) on Minkowski space sliced by non-trivial hypersurfaces:…”

“…From the constraint propagation equations one directly reads off a stability prognosis in the spirit of Frauendiener and Vogel [16]: if χ < 0, a constraint violating continuum instability has to be expected, whereas χ > 0 should result in constraint damping. Both effects will be demonstrated below for a simple class of slices in Minkowski space with a fixed sign of χ.…”