3+1 formalism and bases of numerical relativity - LUTh ...

10.3.3 Constraint-violating modes
10.4 BSSN scheme 181
The constraint preservation property established in the preceding section adds some substantial
support to the concept of free evolution scheme. However this is a mathematical result and it does
not guarantee that numerical solutions will not violate the constraints. Indeed numerical codes
based on free evolution schemes have been plagued for a long time by the so-called constraintviolating
modes. The latter are solutions (γ,K,N,β) which satisfy F = 0 up to numerical
accuracy but with H = 0 and M = 0, although if initially H = 0 and M = 0 (up to numerical
accuracy). The reasons for the appearance of these constraint-violating modes are twofold: (i)
due to numerical errors, the conditions H = 0 and M = 0 are slightly violated in the initial
data, and the evolution equations amplify (in most cases exponentially !) this violation and
(ii) constraint violations may flow into the computational domain from boundary conditions
imposed at timelike boundaries. Notice that the demonstration in Sec. 10.3.2 did not take into
account any boundary and could not rule out (ii).
An impressive amount of works have then been devoted to this issue (see [243] for a review
and Ref. [167, 217] for recent solutions to problem (ii)). We mention hereafter shortly the
symmetric hyperbolic formulations, before discussing the most successful approach to date: the
BSSN scheme.
10.3.4 Symmetric hyperbolic formulations
The idea is to introduce auxiliary variables so that the dynamical equations become a first-order
symmetric hyperbolic system, because these systems are known to be well posed (see e.g. [250,
214]). This comprises the formulation developed in 2001 by Kidder, Scheel and Teukolsky [168]
(KST formulation), which constitutes some generalization of previous formulations developed
by Frittelli and Reula (1996) [137] and by Andersson and York (1999) [13], the latter being
known as the Einstein-Christoffel system.
10.4 BSSN scheme
10.4.1 Introduction
The BSSN scheme is a free evolution scheme for the conformal 3+1 Einstein system (6.105)-
(6.110) which has been devised by Shibata and Nakamura in 1995 [233]. It has been re-analyzed
by Baumgarte and Shapiro in 1999 [43], with a slight modification, and bears since then the
name BSSN for Baumgarte-Shapiro-Shibata-Nakamura.
10.4.2 Expression of the Ricci tensor of the conformal metric
The starting point of the BSSN formulation is the conformal 3+1 Einstein system (6.105)-(6.110).
One then proceeds by expressing the Ricci tensor ˜R of the conformal metric ˜γ, which appears
in Eq. (6.108), in terms of the derivatives of ˜γ. To this aim, we consider the standard expression
of the Ricci tensor in terms of the Christoffel symbols ˜ Γk ij of the metric ˜γ with respect to the