  There is a tendency to write the equations of general relativity as a first
order symmetric system of time dependent partial differential equations.
However, for numerical reasons, it might be advantageous to use a second order
formulation like one obtained from the ADM equations. Unfortunately, the type
of the ADM equations is not well understood and therefore we shall discuss, in
the next section, the concept of wellposedness. We have to distinguish between
weakly and strongly hyperbolic systems. Strongly hyperbolic systems are well
behaved even if we add lower order terms. In contrast; for every weakly
hyperbolic system we can find lower order terms which make the problem totally
illposed. Thus, for weakly hyperbolic systems, there is only a restricted class
of lower order perturbations which do not destroy the wellposedness. To
identify that class can be very difficult, especially for nonlinear
perturbations. In Section 3 we will show that the ADM equations, linearized
around flat with constant lapse function and shift vector, are only weakly
hyperbolic. However, we can use the trace of the metric as a lapse function to
make the equations into a strongly second order hyperbolic system. Using simple
models we shall, in section 4, demonstrate that approximations of second order
equations have better accuracy properties than the corresponding approximations
of first order equations. Also, we avoid spurious waves which travel against
the characteristic direction. In the last section we discuss some difficulties
connected with the preservation of constraints.
