  We numerically analyse the rotation of a neutrally buoyant spheroid in a
shear flow at small shear Reynolds number. Using direct numerical stability
analysis of the coupled nonlinear particle-flow problem we compute the linear
stability of the log-rolling orbit at small shear Reynolds number, ${\rm
Re}_a$. As ${\rm Re}_a \to 0$ and as the box size of the system tends to
infinity we find good agreement between the numerical results and earlier
analytical predictions valid to linear order in ${\rm Re}_a$ for the case of an
unbounded shear. The numerical stability analysis indicates that there are
substantial finite-size corrections to the analytical results obtained for the
unbounded system. We also compare the analytical results to results of
lattice-Boltzmann simulations to analyse the stability of the tumbling orbit at
shear Reynolds numbers of order unity. Theory for an unbounded system at
infinitesimal shear Reynolds number predicts a bifurcation of the tumbling
orbit at aspect ratio $\lambda_{\rm c} \approx 0.137$ below which tumbling is
stable (as well as log rolling). The simulation results show a bifurcation line
in the $\lambda$-${\rm Re}_a$ plane that reaches $\lambda \approx0.1275$ at the
smallest shear Reynolds number (${\rm Re}_a=1$) at which we could simulate with
the lattice-Boltzmann code, in qualitative agreement with the analytical
results.
