(ed.). Gravitational waves (IOP, 2001)(422s).

368 Numerical relativityinformation propagating with a finite speed, the system does not form a hyperbolicsystem, and is not necessarily the best for numerical evolution. Other fieldsof physics, in particular hydrodynamics, have developed very mature numericalmethods that are specially designed to treat the well studied flux conservative,hyperbolic system of balance laws having the form∂ t u + ∂ k F k − u = S − u (18.11)where the vector u displays the set of variables and both ‘fluxes’ F k and ‘sources’S are vector valued functions. In hydrodynamic systems, it often turns out thatthe characteristic matrix ∂ F/∂ u projected into any spacelike direction can oftenbe diagonalized, so that fields with definite propagation speeds can be identified(the eigenvectors and the eigenvalues of the projected characteristic matrix). Oneimportant point is that in (18.11) all spatial derivatives are contained in the fluxterms, with the source terms in the equations containing no derivatives of theeigenfields. All of these features can be exploited in numerical finite differenceschemes that treat each term in an appropriate way to preserve important physicalcharacteristics of the solution.Amazingly, the complete set of Einstein equations can also be put inthis ‘simple’ form (the source terms still contain thousands of terms however).Building on earlier work by Choquet-Bruhat and Ruggeri [30], Bona and Massóbegan to study this problem in the late 1980s, and by 1992 they had developeda hyperbolic system for the Einstein equations with a certain specific gaugechoice [31] (see below). Here by hyperbolic, we mean simply that the projectedcharacteristic matrix has a complete set of eigenfields with real eigenvalues.This work was generalized recently to apply to a large family of gauge choices[32, 33]. The Bona–Massó system of equations is available in the 3D ‘Cactus’code [34,35], as is the standard ADM system, where both are tested and comparedon a number of spacetimes.The Bona–Massó system is now one among many hyperbolic systems,as other independent hyperbolic formulations of Einstein’s equations weredeveloped [36–41] at about the same time as [42]. Among these otherformulations only the one originally devised in [38] has been applied tospacetimes containing black holes [43], although still only in the sphericallysymmetry 1D case (a 3D version is under development [44].) Hence, of themany hyperbolic variants, only the Bona–Massó family and the formulationsof York and co-workers have been tested in any detail in 3D numerical codes.Notably among the differences in the formulations, the Bona–Massó and Frittellifamilies contain terms equivalent to second time derivatives of the three metricγ ab , while many other formulations go to a higher time derivative to achievehyperbolicity. Another comment worth making is that for harmonic slicing, boththe Bona–Massó and York families have characteristic speeds of either zero, orlight speed. For maximal slicing, they both reduce to a coupled elliptic-hyperbolicsystem. The Bona–Massó system (at least) also allows for an additional familyof explicit algebraic slicings, with the lapse proportional to an explicit function