By Harrison D.M.

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These evolution equations have been presented in what is essentially the "standard form" , which has served numerical relativity well over the last few decades. However, the equations are enormously complicated, and their form is not necessarily the best for numerical evolution. Other fields of physics, such as hydrodynamics, have developed very mature numerical methods that are specially designed to treat various terms in the equations. Recently there have been significant advances in casting the Einstein equations in a first-order, 30 Edward Seidel flux-conservative, hyperbolic (FOFCH) form for which advanced numerical methods can be borrowed from hydrodynamics for the first time [11,12] (see also recent work in [13,14]).

This term may be obtained by using the extrapolation formula K n+l/ 2 i,j,k _ - ~Kn 2 i,j,k _ ~Kn-l 2 i,j,k' (20) so (19) defines K n +1/2 only in terms of known quantities. This method has been used in previous codes and was found to work well [23,24,29]. Alternatively, one can also use a Taylor expansion of the evolution equations to get n + 1 / 2 , which has more mathematically motivated expansions for terms like K 't,), k been shown to work better in some applications (Anninos, P. (1994): private communication) .

Another approach to evolution involves so-called "symplectic" schemes, developed by Berger and Moncrief (as discussed by Berger in this book) in the context of numerical relativity. These schemes use a Hamiltonian approach to generate the time-evolution operator, and are discussed in detail by Berger in this book. This is an example of another powerful new approach for evolving the Einstein equations that may play an important role in future studies. If the constraints are satisfied on any hypersurface (as they will be if the initial-value problem has been solved properly), the Bianchi identities then guarantee that they remain satisfied on all subsequent hypersurfaces.

### About Mass-Energy Equivalence by Harrison D.M.

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