Adaptive Smoothed Particle Hydrodynamics, with Application to Cosmology: Methodology

The development of a new smoothed particle hydrodynamics (SPH) method, called adaptive smoothed particle hydrodynamics (ASPH), generalized for cosmology and coupled to the particle mesh (PM) method for solving the Poisson equation, for the gasdynamical simulation of galaxy and large-scale structure formation, will be described. The accurate numerical simulation of the highly nonlinear phenomena of shocks and caustics which occur generically in the process of cosmological structure formation requires enormous dynamic range and resolution. Existing numerical methods require substantial modification in order to achieve the required resolution with current computer technology. The SPH method is a promising approach to this problem, since it is a Lagrangian numerical hydrodynamics method which adjusts its resolution dynamically so as to keep track of the mass as it flows. However, in its standard form, SPH suffers from two deficiencies which are particularly acute in the presence of the kind of gravitational collapse and strong shocks which are generic to the dynamics of galaxy and large-scale structure formation. The first deficiency results from the fact that the smoothing kernel in standard SPH is isotropic, while gravitational collapse and shock waves involve highly anisotropic volume changes. Hence, the ability of SPH to adjust its resolution dynamically so as to follow Lagrangian fluid changes is limited by the mismatch between this isotropic smoothing kernel and the inherent anisotropy of the flow. The ASPH method solves this problem by replacing the isotropic smoothing kernel of standard SPH, which is characterized by a scalar smoothing length h, by an an isotropic smoothing tensor H which adjusts dynamically so as to follow the changes of the local mean interparticle spacing with direction around each fluid element. The second deficiency of standard SPH results from the fact that artificial viscosity is necessary in order to accommodate shocks, but this results in substantial and widespread artificial heating of gas which is undergoing supersonic collapse far from any shock. The ASPH method solves this problem by using the evolution of the anisotropic smoothing tensor II to track shocks by predicting the occurrence of caustics in the flow, and thereby to restrict the effect of artificial viscosity to just those fluid particles which are involved in the shock transition. The result of these two new features introduced by ASPH is a substantial increase in the resolving power of the SPH method for the same total number of particles, without any corresponding increase in computational run time. The new algorithms which constitute the ASPH method are described here in detail. A series of tests of the method in one and two dimensions are presented. These include kinematical tests of the anisotropic smoothing algorithm and a comparison with that of standard SPH, as well as dynamical tests, including the Reimann shock tube problem. A special emphasis is placed on the requirement that the method pass the stringent test of matching the known, detailed solution of the cosmological pancake collapse problem. Finally, we apply the ASPH method in two dimensions to simulate the growth of large-scale structure from a spectrum of primordial density fluctuations in a hot dark matter model. The ASPH method succeeds in resolving the generic nonlinear structures and shocks in such a model in a calculation with fewer than 40 particles per pancake per dimension.