  Accurate simulations of flows in stellar interiors are crucial to improving
our understanding of stellar structure and evolution. Because the typically
slow flows are merely tiny perturbations on top of a close balance between
gravity and the pressure gradient, such simulations place heavy demands on
numerical hydrodynamics schemes. We demonstrate how discretization errors on
grids of reasonable size can lead to spurious flows orders of magnitude faster
than the physical flow. Well-balanced numerical schemes can deal with this
problem. Three such schemes were applied in the implicit, finite-volume
Seven-League Hydro (SLH) code in combination with a low-Mach-number numerical
flux function. We compare how the schemes perform in four numerical experiments
addressing some of the challenges imposed by typical problems in stellar
hydrodynamics. We find that the $\alpha$-$\beta$ and deviation well-balancing
methods can accurately maintain hydrostatic solutions provided that
gravitational potential energy is included in the total energy balance. They
accurately conserve minuscule entropy fluctuations advected in an isentropic
stratification, which enables the methods to reproduce the expected scaling of
convective flow speed with the heating rate. The deviation method also
substantially increases accuracy of maintaining stationary orbital motions in a
Keplerian disk on long timescales. The Cargo-LeRoux method fares substantially
worse in our tests, although its simplicity may still offer some merits in
certain situations. Overall, we find the well-balanced treatment of gravity in
combination with low Mach number flux functions essential to reproducing
correct physical solutions to challenging stellar slow-flow problems on
affordable collocated grids.
