9 FIG. 4: Convergence results for the evolution of a stable, nonmagnetized, rotating star (the particular initial solution is shown in Fig. 2 as a green solid square). Three FMR evolutions are shown, each with a multiple of the coarsest resolution resolving the equator of the star with: (magenta, dot-dashed) 30 points/star, (blue, dotted) 60 points/star, and (black, solid) 120 points/star. Also shown is a run with the same stellar resolution of (red, long-dashed) 60 points/star but with a coarse grid that extends twice as far in all direc- tions. The two highest resolution runs were terminated early because of the computational cost, not because of any robust- ness problems. The insets show the same data but in finer detail for the first rotational period. The top frame shows the fractional change in maximum density versus rotational period. With increasing resolution the solution better ap- proximates a stationary solution. The upper middle frame shows the fractional change in the baryon mass, Although our scheme, using vertex centered AMR, an atmosphere, and full general relativity with sources, is not strictly conservative, the plot shows that deviations from strict conservation are small and decrease with more resolution. The lower middle frame shows the fractional change in integrated angular momentum. Again, the code demonstrates convergence to conservation. The bottom frame shows the maximum of the norm of the Z a constraint residuals as a function of time which also con- verge. excited by inherent numerical error. However, the figure shows data for a number of resolutions, and the trend as resolution improves is toward a flatter curve. This trend is particularly apparent in the inset which shows just the first rotational period. This behavior suggests that the code is converging to the continuum solution. In addition to the runs with varying resolution, Fig. 4 shows another evolution with a coarse level extending Similarly, we evolve a rotating, magnetized star located on the unstable side. These stars, perturbed by inherent numerical error, collapse to black holes. We see no evi- dence of significant disk formation (as in e.g. [15, 16, 54– Finally, the bottom frame of Fig. 4 shows the maxi- mum value of the constraint violation. These values in- crease with time as the numerical error accumulates, but higher resolution runs demonstrate less violation though it saturates due to the intrinsic error of the initial and boundary data. Although the fluid scheme used here is not strictly con- servative because of (i) AMR boundaries for our vertex centered scheme, (ii) our outer boundary treatment, and (iii) our use of a fluid floor, Fig. 4 shows that the frac- tional change in the volume-integrated baryon mass re- mains constant to a high degree. Similarly, the integrated angular momentum of the fluid converges to conserva- tion. twice as far but with fine levels identical to the medium resolution run just discussed. Generally, the results are the same as for the non-extended domain, indicating little effect from the boundary for the the first half-period. FIG. 5: Collapse to black hole of an unstable, unperturbed, magnetized star for multiple of a base resolution. The ini- tial data is an unperturbed Magstar solution with central en- thalpy H$_{c}$ = 0 . 8 rotating at a frequency f = 835 Hz and with polar magnetic field of 1000 GT = 10 16 G . The top frame shows the maximum density which increases with time as the star collapses. The upper middle frame shows a norm of the divergence of the magnetic field. The lower frames show the distortion parameters of the D field as functions of time. Unstable, Rotating, Magnetized Star Test: