  Theoretical studies of magnets have traditionally concentrated on either
classical spins, or the extreme quantum limit of spin-1/2. However, magnets
built of spin-1 moments are also intrinsically interesting, not least because
they can support quadrupole, as well as dipole moments, on a single site. For
this reason, spin-1 models have been extensively studied as prototypes for
quadrupolar (spin-nematic) order in magnetic insulators, and Fe-based
superconductors. At the same time, because of the presence of quadrupoles, the
classical limit of a spin-1 moment is not an $O(3)$ vector, a fact which must
be taken into account in describing their properties. In this Article we
develop a method to simulate spin-1 magnets based on a $u(3)$ algebra which
treats both dipole and quadrupole moments on equal footing. This approach is
amenable to both classical and quantum calculations, and we develop the
techniques needed to calculate thermodynamic properties through Monte Carlo
simulations and classical low-temperature expansion, and dynamical properties,
through "molecular dynamics" simulations and a multiple-boson expansion. As a
case study, we present detailed analytic and numerical results for the
thermodynamic properties of ferroquadrupolar order on the triangular lattice,
and its associated dynamics. At low temperatures, we show that it is possible
to "correct" for the effects of classical statistics in simulations, and
extrapolate to the zero-temperature quantum results found in flavour-wave
theory.
