100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

428 R. Gambini and J. Pullin0.020.015∆=5.097e-3∆=5.10107e-4∆=2.45e-4(O 2-O 2exact)/O20.010.00500 100 200 300 400 500nFig. 6. Convergence of the method with increasing resolution. We display the relativeerror in one of the perennials of the theory as a function of n. The two runs with thecoarser resolutions are shown with one point out of every ten displayed. The finer runis shown with one point displayed out of every thirty. The range of n displayed correspondsto a full trajectory along the ellipse in configuration space. So the improvementin accuracy is throughout the entire evolution with different levels of improvement atdifferent points. See the text as to why we fine tune the evolution steps for the variousruns in the convergence study.step smaller lowers the errors. However, one notes differences with the usualtype of convergence in the sense that here we have that it is not uniform asa function of the evolution time. One notes, for example, that at isolatedpoints some of the coarser runs have very low errors. To understand thisone needs to recall that in our approach discrete expressions differ from thecontinuum ones as,O = O continuum + f(p, q)C (36)with C the constraints. It could happen that at a particular point in a coarseevolution the constraint chances to take a value very close to zero. Slightlyfiner resolution runs may not land on top of that particular point andtherefore will apparently have larger errors in the region. Eventually, if one