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A two-level nested grid code is applied to resolve X-ray clusters in a standard critically closed cold dark matter dominated universe. The physical dimension of the larger periodically identified box is set to $50~Mpc$ as a compromise to providing both adequate sampling of long wavelength perturbations and sufficient small scale resolution. A refined grid with smaller cell dimensions is constructed within the larger cube to resolve a single rich cluster in greater detail as the larger scale structure evolves on the parent grid. We performed a sequence of runs at consistently higher resolution to test for the convergence of the various physical attributes of X-ray clusters, including the X-ray luminosity, the Sunyaev-Zel'dovich decrement and the $\beta$-model parameters. Although we find some evidence of a convergence pattern in the cluster core radius, we are unable to converge on the integrated X-ray luminosity (a quantity that is especially sensitive to the baryonic density in the inner core of the cluster) even at the most refined subgrid resolution of $100~kpc$, effectively a $512^3$ grid covering the cluster proper. Averaged radial profiles of the density (gas and dark matter) and gas temperature are consistent with Bertschinger's (1985) self-similar solution at small radii (to the force softening length), which yields a divergent total luminosity function. We also investigate the reliability of reconstructing the numerical data based solely on the ``observed'' X-ray luminosities and the isothermal and hydrostatic equilibrium assumptions made in the standard $\beta$-models. This work is carried out in a developing framework to explore the advantages and limitations of nested grid methods as applied to cosmological simulations.
