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

Still newer formulations: towards a stable evolution system 373split Lax-Wendroff method have been implemented and tested extensively in‘Cactus’. These schemes are fully second order in space and time. Shockcapturing methods have been shown to work extremely well in 1D problemsin numerical relativity [32, 70], but their application in 3D is an active researcharea full of promise, but as yet, unfulfilled. The details of these methods, asthey are applied to the Bona–Massó formulation of the equations, can be foundin [34, 70].18.3.0.2 The role of constraintsIf the constraints are satisfied on the initial hypersurface, the evolution equationsthen guarantee that they remain satisfied on all subsequent hypersurfaces. Thus,once the initial value problem has been solved, one may advance the solutionforward in time by using only the evolution equations. This is the same situationencountered in electrodynamics as discussed before. However, in a numericalsolution, the constraints will be violated at some level due to numerical error.They hence provide useful indicators for the accuracy of the numerical spacetimesgenerated. Traditional alternatives to this approach, which is often referred toas ‘free evolution’, involve solving some or all of the constraint equations oneach slice for certain metric and extrinsic curvature components, and then simplymonitoring the ‘left over’ evolution equations. This issue is discussed further byChoptuik in [72], and in detail for the Schwarzschild spacetime in [73]. Newapproaches to this problem of constraint versus evolution equations are currentlybeing pursued by Lee [74], among others [75]. This approach is to advancethe system forward using the evolution equations, and then adjust the variablesslightly so that the constraints are satisfied (to some tolerance), i.e. the solutionis projected onto the constraint surface. Because there are many variables thatgo into the constraints, there is not a unique way to decide which ones to adjustand by how much. However, one can compute the ‘minimum’ perturbation tothe system, which corresponds to projecting to the closest point on the constraintsurface. Other approaches, similar in spirit to each other, have been suggestedby Detweiler [76] and Brodbeck et al [77]. The Detweiler approach restricts thenumerical evolution to the constraint surface by adding terms to the evolutionequations (18.9) and (18.10) terms which are proportional to the constraints.Numerical tests of the scheme using gravitational wave spacetimes have recentlybeen carried out, showing promising results [78].18.3.0.3 Gauge conditionsKinematic conditions for the lapse function α and shift vector β i have to bespecified for the evolution equations (18.9) and (18.10). With γ ab and K absatisfying the constraint on the initial slice, the lapse and shift can be chosenarbitrarily on the initial slice and thereafter. These are referred to as gaugechoices, analogous to the choice of the gauge function in electrodynamics.