Consistency of finite-difference solutions of Einstein’s equations

In the past, arguments have been advanced suggesting that certain finite-difference solutions of the 3+1 form of Einstein's equations suffer from a fundamental inconsistency. Specifically, it has been claimed that freely evolved solutions, where the constraint equations are not explicitly imposed after the initial time, will generally satisfy discrete versions of the constraints to a lower order in the basic mesh spacing h than the truncation order of the discretized evolution equations. This issue is reexamined here, and using the key observation, originally due to Richardson, that a numerical differentiation need not produce an O (h^p-1) quantity from an O (h^p) one, it is argued that there should be no such inconsistency for convergent difference schemes. Numerical results from a study of spherically symmetric solutions of a massless scalar field minimally coupled to the gravitational field are presented in support of this claim. These results show that the expected convergence of various residual quantities can be achieved in practice.