  Self-Interacting Dark Matter (SIDM) is a collisional form of cold dark matter
(CDM), originally proposed to solve problems that arose when the collisionless
CDM theory of structure formation was compared with observations of galaxies on
small scales. The quantitative impact of the proposed elastic collisions on
structure formation has been estimated previously by Monte Carlo N-body
simulations and by a conducting fluid model, with apparently diverging results.
To improve this situation, we make direct comparisons between new Monte Carlo
N-body simulations and solutions of the conducting fluid model, for isolated
SIDM haloes of fixed mass. This allows us to separate cleanly the effects of
gravothermal relaxation from those of continuous mass accretion in an expanding
background universe. When these two methods were previously applied to halo
formation with cosmological boundary conditions, they disagreed by an order of
magnitude about the size of the scattering cross section required to solve the
so-called 'cusp-core problem.' We show here, however, that the methods agree
with each other within 20 per cent for isolated haloes. This suggests that the
two methods are consistent, and that their disagreement for cosmological haloes
is not caused by a breakdown of their validity.
  The isolated haloes studied here undergo gravothermal collapse. We compare
the solutions calculated by these two methods for gravothermal collapse
starting from several initial conditions. This allows us to calibrate the heat
conduction which accounts for the effect of elastic hard-sphere scattering in
the fluid model. The amount of tuning of the thermal conductivity parameters
required to bring the two methods into close agreement for isolated haloes,
however, is too small to explain the discrepancy found previously in the
cosmological context.
