  Hydrodynamic interactions in systems comprised of self-propelled particles,
such as swimming microorganisms, and passive tracers have a significant impact
on the tracer dynamics compared to the equivalent "dry" sample. However, such
interactions are often difficult to take into account in simulations due to
their computational cost. Here, we perform a systematic investigation of
swimmer-tracer interaction using an efficient force/counter-force based
lattice-Boltzmann (LB) algorithm [J. de Graaf~\textit{et al.}, J. Chem.
Phys.~\textbf{144}, 134106 (2016)] in order to validate its ability to capture
the relevant low-Reynolds-number physics. We show that the LB algorithm
reproduces far-field theoretical results well, both in a system with periodic
boundary conditions and in a spherical cavity with no-slip walls, for which we
derive expressions here. The force-lattice coupling of the LB algorithm leads
to a "smearing out" of the flow field, which strongly perturbs the tracer
trajectories at close swimmer-tracer separations, and we analyze how this
effect can be accurately captured using a simple renormalized hydrodynamic
theory. Finally, we show that care must be taken when using LB algorithms to
simulate systems of self-propelled particles, since its finite momentum
transport time can lead to significant deviations from theoretical predictions
based on Stokes flow. These insights should prove relevant to the future study
of large-scale microswimmer suspensions using these methods.
