We present a comprehensive set of convergence tests which explore the role of various numerical parameters on the equilibrium structure of a simulated dark matter halo .
We report results obtained with two independent , state-of-the-art , multi-stepping , parallel N–body codes : PKDGRAV and GADGET .
We find that convergent mass profiles can be obtained for suitable choices of the gravitational softening , timestep , force accuracy , initial redshift , and particle number .
For softenings chosen so that particle discreteness effects are negligible , convergence in the circular velocity is obtained at radii where the following conditions are satisfied : ( i ) the timestep is much shorter than the local orbital timescale ; ( ii ) accelerations do not exceed a characteristic acceleration imprinted by the gravitational softening ; and ( iii ) enough particles are enclosed so that the collisional relaxation timescale is longer than the age of the universe .
Convergence also requires sufficiently high initial redshift and accurate force computations .
Poor spatial , time , or force resolution leads generally to systems with artificially low central density , but may also result in the formation of artificially dense central cusps .
We have explored several adaptive time-stepping choices and obtained best results when individual timesteps are chosen according to the local acceleration and the gravitational softening ( \Delta t _ { i } \propto ( \epsilon / a _ { i } ) ^ { 1 / 2 } ) , although further experimentation may yield better and more efficient criteria .
The most stringent requirement for convergence is typically that imposed on the particle number by the collisional relaxation criterion , which implies that in order to estimate accurate circular velocities at radii where the density contrast may reach \sim 10 ^ { 6 } , the region must enclose of order 3000 particles ( or more than a few times 10 ^ { 6 } within the virial radius ) .
Applying these criteria to a galaxy-sized \Lambda CDM halo , we find that the spherically-averaged density profile becomes progressively shallower from the virial radius inwards , reaching a logarithmic slope shallower than -1.2 at the innermost resolved point , r \sim 0.005 r _ { 200 } , with little evidence for convergence to a power-law behaviour in the inner regions .