Constraint and gauge shocks in one-dimensional numerical relativity

Abstract

We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can be expected. One criteria is related with the so-called geometric blow-up leading to gradient catastrophes, while the other is based upon the ODE-mechanism leading to blow-ups within finite time. We show how both mechanisms work in the case of a simple one-dimensional wave equation with a dynamic wave speed and sources, and later explore how those blow-ups can appear in one-dimensional numerical relativity. In the latter case we recover the well known "gauge shocks" associated with Bona-Masso type slicing conditions. However, a crucial result of this study has been the identification of a second family of blow-ups associated with the way in which the constraints have been used to construct a hyperbolic formulation. We call these blow-ups "constraint shocks" and show that they are formulation specific, and that choices can be made to eliminate them or at least make them less severe.

Generalized harmonic spatial coordinates and hyperbolic shift conditions

“…As pointed out in [26], the c iii w 2 i component of the source term can be expected to dominate and to cause blowups in the solution within a finite time. In order to avoid these blowups we therefore demand that the coefficients c iii should vanish, and we refer to this condition as the ''source criteria.''…”

“…In [26] some of us presented blowup avoiding conditions for both these mechanisms, which we called ''indirect linear degeneracy'' [25] and the ''source criteria.'' In that reference it was also shown, using numerical examples, that the source criteria for avoiding blowups is generally the more important of the two conditions.…”

“…A criteria to avoid such blowups for systems of partial differential equations of the form (5.1) and (5.2) was proposed by some of us in [26]: When diagonalizing the evolution system for the v's, making use of (5.3) and (5.4), one finds…”

“…7 (which should be compared with Fig. 7 of [26]) shows the rms of the Hamiltonian constraint at time t 100, as a function of the adjustment parameters h K and h K B . We can see that the line (7.33) obtained by the source criteria is numerically preferred, although there does seem to be a discrepancy for large values of h K B for which the numerical results suggest a somewhat steeper line.…”

Continuum and Discrete Initial-Boundary Value Problems and Einstein’s Field Equations

“…In general, shocks may form when the system is not linearly degenerated or genuinely nonlinear [250]. The Einstein vacuum equations, on the other hand, can be written in linearly degenerate form (see, for example, [6, 7, 348, 8]) and are therefore expected to be free of physical shocks. For these reasons, one cannot expect global existence of smooth solutions from smooth initial data with compact support in general, and the best one can hope for is existence of a smooth solution on some finite time interval [0, T ], where T might depend on the initial data.…”