  This paper presents an alternative formulation of the ASPH algorithm for
evolving anisotropic smoothing kernels, in which the geometric approach of
Shapiro et al. (1996; Paper I) is replaced by an approach involving a local
transformation of coordinates to those in which the underlying anisotropic
volume changes appear to be isotropic. The ASPH method is presented in 2D and
3D, including a number of details not previously included in Paper I, some of
which represent either advances or different choices with respect to Paper I.
Among the advances included here are an asynchronous time-integration scheme
with different time steps for different particles and the generalization of the
ASPH method to 3D. The shock-tracking algorithm described in Paper I for
locally adapting the artificial viscosity to restrict viscous heating just to
particles encountering shocks, is not included here. Instead, we adopt a
different interpolation kernel for use with the artificial viscosity, which has
the effect of spatially localizing effects of the artificial viscosity. This
version of the ASPH method in 2D and 3D is then applied to a series of 1D, 2D,
and 3D test problems, and the results are compared to those of standard SPH
applied to the same problems. These include the problem of cosmological pancake
collapse, the Riemann shock tube, cylindrical and spherical Sedov blast waves,
the collision of two strong shocks, and problems involving shearing disks
intended to test the angular momentum conservation properties of the method.
These results further support the idea that ASPH has significantly better
resolving power than standard SPH for a wide range of problems, including that
of cosmological structure formation. (Abridged)
