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

372 Numerical relativityHyperbolic evolutionsThe hyperbolic formulations are on a much firmer footing numerically than theADM formulation, as the equations are in a much simpler form that has beenstudied for many years in computational fluid dynamics. However, the applicationof such methods to relativity is quite new, and hence the experience with suchmethods in this community is relatively limited. Furthermore, the treatment of thehighly nonlinear source terms that arise in relativity is very much unexplored, andthe source terms in Einstein’s equations are much more complicated than those inhydrodynamics. Here we will just discuss the basic ideas in such schemes.A standard technique for equations having flux conservative form is to splitequation (18.11) into two separate processes. The transport part is given by theflux terms∂ t u + ∂ k F− k u = 0. (18.12)The source contribution is given by the following system of ordinary differentialequations∂ t u = S − u. (18.13)Numerically, this splitting is performed by a combination of both flux and sourceoperators. Denoting by E(t) the numerical evolution operator for system(18.11) in a single timestep, we implement the following combination sequenceof subevolution steps:E(t) = S(t/2)T (t)S(t/2) (18.14)where T , S are the numerical evolution operators for systems (18.12) and (18.13),respectively. This is known as ‘Strang splitting’ [66]. As long as both operatorsT and S are second-order accurate in t, the overall step of operator E is alsosecond-order accurate in time.This choice of splitting allows easy implementation of different numericaltreatments of the principal part of the system without having to worry aboutthe sources of the equations. Additionally, there are numerous computationaladvantages to this technique, as discussed in [67].The sources can be updated using a variety of ODE integrators, and in‘Cactus’ the usual technique involves second-order predictor-corrector methods.Higher order methods for source integration can be easily implemented, but thiswill not improve the overall order of accuracy. However, in special cases wherethe evolution is largely source driven [68], it may be important to use higher ordersource operators, and this method allows such generalizations. The details can befound in [34].The implementation of numerical methods for the flux operator is muchmore involved, and there are many possibilities, ranging from standard choicesto advanced shock capturing methods [33, 69, 70]. Among standard methods, theMacCormack method, which has proven to be very robust in the computationalfluid dynamics field (see, e.g., [71] and references therein), and a directionally