  Many complex fluids can be described by continuum hydrodynamic field
equations, to which noise must be added in order to capture thermal
fluctuations. In almost all cases, the resulting coarse-grained stochastic
partial differential equations carry a short-scale cutoff -- which is also
reflected in numerical discretisation schemes. We draw together our recent
findings concerning the construction of such schemes and the interpretation of
their continuum limits, focusing for simplicity on models with a purely
diffusive scalar field, such as `Model B' which describes phase separation in
binary fluid mixtures. We address the requirement that the steady state entropy
production rate (EPR) must vanish for any stochastic hydrodynamic model in
thermal equilibrium. Only if this is achieved can the given discretisation
scheme be relied upon to correctly calculate the nonvanishing EPR for `active
field theories' in which new terms are deliberately added to the fluctuating
hydrodynamic equations that break detailed balance. To compute the correct
probabilities of forward and time-reversed paths (whose ratio determines the
EPR) we must make a careful treatment of so-called `spurious drift' and other
closely related terms that depend on the discretisation scheme. We show that
such subtleties can arise not only in the temporal discretisation (as is well
documented for stochastic ODEs with multiplicative noise) but also from spatial
discretisation, even when noise is additive, as most active field theories
assume. We then review how such noise can become multiplicative, via
off-diagonal couplings to additional fields that encode thermodynamically the
underlying chemical processes responsible for activity. In this case the
spurious drift terms need careful accounting, not just to evaluate correctly
the EPR, but also to numerically implement the Langevin dynamics itself.
