Consistency of finite-difference solutions of Einstein's equations

Details Consistency of finite-difference solutions of Einstein's equations Consistency of finite-difference solutions of Einstein's equations

Nov 1991

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991phrvd..44.3124c&link_type=abstract

Physical Review D (Particles, Fields, Gravitation, and Cosmology), Volume 44, Issue 10, 15 November 1991, pp.3124-3135

Physics

49

Exact Solutions

Scientific paper

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(hp-1) quantity from an O(hp) 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.

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