  In simulating continuum model fluids that undergo phase separation and
criticality, significant gains in computational efficiency may be had by
confining the particles to the sites of a lattice of sufficiently fine spacing,
$a_{0}$ (relative to the particle size, say $a$). But a cardinal question,
investigated here, then arises, namely: How does the choice of the lattice
discretization parameter, $\zeta\equiv a/a_{0}$, affect the values of
interesting parameters, specifically, critical temperature and density,
$T_{\scriptsize c}$ and $\rho_{\scriptsize c}$? Indeed, for small $\zeta
(\lesssim 4 $-$ 8)$ the underlying lattice can strongly influence the
thermodynamic properties. A heuristic argument, essentially exact in $d=1$ and
$d=2$ dimensions, indicates that for models with hard-core potentials, both
$T_{\scriptsize c}(\zeta)$ and $\rho_{\scriptsize c}(\zeta)$ should converge to
their continuum limits as $1/\zeta^{(d+1)/2}$ for $d\leq 3$ when
$\zeta\to\infty$; but the behavior of the error is highly erratic for $d\geq
2$. For smoother interaction potentials, the convergence is faster. Exact
results for $d=1$ models of van der Waals character confirm this; however, an
optimal choice of $\zeta$ can improve the rate of convergence by a factor
$1/\zeta$. For $d\geq 2$ models, the convergence of the {\em second virial
coefficients} to their continuum limits likewise exhibit erratic behavior which
is seen to transfer similarly to $T_{\scriptsize c}$ and $\rho_{\scriptsize
c}$; but this can be used in various ways to enhance convergence and improve
extrapolation to $\zeta = \infty$ as is illustrated using data for the
restricted primitive model electrolyte.
