ADAPTIVE SMOOTHED PARTICLE HYDRODYNAMICS - METHODOLOGY - II

Further development and additional details and tests of adaptive smoot hed particle hydrodynamics (ASPH), the new version of smoothed particl e hydrodynamics (SPH) described in the first paper in this series (Sha piro et al.), are presented. The ASPH method replaces the isotropic sm oothing algorithm of standard SPH, in which interpolation is performed with spherical kernels of radius given by a scalar smoothing length, with anisotropic smoothing involving ellipsoidal kernels and tensor sm oothing lengths. In standard SPH, the smoothing length for each partic le represents the spatial resolution scale in the vicinity of that par ticle and is typically allowed to vary in space and time so as to refl ect the local value of the mean interparticle spacing. This isotropic approach is not optimal, however, in the presence of strongly anisotro pic volume changes such as occur naturally in a wide range of astrophy sical flows, including gravitational collapse, cosmological structure formation, cloud-cloud collisions, and radiative shocks. In such cases , the local mean interparticle spacing varies not only in time and spa ce but also in direction as well. This problem is remedied in ASPH, wh ere each axis of the ellipsoidal smoothing kernel for a given particle is adjusted so as to reflect the different mean interparticle spacing s along different directions in the vicinity of that particle. By defo rming and rotating these ellipsoidal kernels so as to follow the aniso tropy of volume changes local to each particle, ASPH adapts its spatia l resolution scale in time, space, and direction. This significantly i mproves the spatial resolving power of the method over that of standar d SPH at fixed particle number per simulation. This paper presents an alternative formulation of the ASPH algorithm for evolving anisotropic smoothing kernels, in which the geometric approach of the first paper in this series, based upon the Lagrangian deformation of ellipsoidal fluid elements surrounding each particle, is replaced by an approach i nvolving a local transformation of coordinates to those in which the u nderlying anisotropic volume changes appear to be isotropic. Using thi s formulation the ASPH method is presented in two and three dimensions , including a number of details not previously included in the earlier paper, some of which represent either advances or different choices w ith respect to the ASPH method detailed in the earlier paper. Among th e advances included here are an asynchronous time-integration scheme w ith different time steps for different particles and the generalizatio n of the ASPH method to three dimensions. In the category of different choices, the shock-tracking algorithm described in the earlier paper for locally adapting the artificial viscosity to restrict viscous heat ing just to particles encountering shocks is not included here. Instea d, we adopt a different interpolation kernel for use with the artifici al viscosity, which has the effect of spatially localizing effects of the artificial viscosity. This version of the ASPH method in two and t hree dimensions is then applied to a series of one-, two-, and three-d imensional test problems, and the results are compared to those of sta ndard SPH applied to the same problems. These include the problem of c osmological pancake collapse, the Riemann shock tube, cylindrical and spherical Sedov blast waves, the collision of two strong shocks, and p roblems 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 stand ard SPH for a wide range of problems, including that of cosmological s tructure formation.