The standard formulation of the smoothed particle hydrodynamics (SPH) assumes that the local density distribution is differentiable. This assumption is used to derive the spatial derivatives of other quantities. However, this assumption breaks down a
t the contact discontinuity. At the contact discontinuity, the density of the low-density side is overestimated while that of the high-density side is underestimated. As a result, the pressure of the low (high) density side is over (under) estimated. Thus, unphysical repulsive force appears at the contact discontinuity, resulting in the effective surface tension. This tension suppresses fluid instabilities. In this paper, we present a new formulation of SPH, which does not require the differentiability of density. Instead of the mass density, we adopt the internal energy density (pressure), and its arbitrary function, which are smoothed quantities at the contact discontinuity, as the volume element used for the kernel integration. We call this new formulation density independent SPH (DISPH). It handles the contact discontinuity without numerical problems. The results of standard tests such as the shock tube, Kelvin-Helmholtz and Rayleigh-Taylor instabilities, point like explosion, and blob tests are all very favorable to DISPH. We conclude that DISPH solved most of known difficulties of the standard SPH, without introducing additional numerical diffusion or breaking the exact force symmetry or energy conservation. Our new SPH includes the formulation proposed by Ritchie & Thomas (2001) as a special case. Our formulation can be extended to handle a non-ideal gas easily.
We propose a symmetrized form of the softened gravitational potential which is a natural extension of the Plummer potential. The gravitational potential at the position of particle i (x_i,y_i,z_i), induced by particle j at (x_j,y_j,z_j), is given by:
phi_ij = -G m_j/|r_ij^2+e_i^2+e_j^2|^1/2, where G is the gravitational constant, m_j is the mass of particle j, r_ij = |(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2|^1/2 and e_i and e_j are the gravitational softening lengths of particles i and j, respectively. This form is formally an extension of the Newtonian potential to five dimensions. The derivative of this equation in the x,y, and z directions correspond to the gravitational accelerations in these directions and these are always symmetric between two particles. When one applies this potential to a group of particles with different softening lengths, as is the case with a tree code, an averaged gravitational softening length for the group can be used. We find that the most suitable averaged softening length for a group of particles is <e_j^2> = sum_j^N m_j e_j^2 / M, where M = sum_j^N m_j and N are the mass and number of all particles in the group, respectively. The leading error related to the softening length is O(sum_j r_j d(e_j^2)/r_ij^3), where r_j is the distance between particle j and the center of mass of the group and d(e_j^2) = e_j^2 - <e_j^2>. Using this averaged gravitational softening length with the tree method, one can use a single tree to evaluate the gravitational forces for a system of particles with a wide variety of gravitational softening lengths. Consequently, this will reduce the calculation cost of the gravitational force for such a system with different softenings without the need for complicated forms of softening. We present the result of simple numerical tests. We found that our modification of the Plummer potential works well.
We describe a new algorithm for the integration of self-gravitating fluid systems using SPH method. We split the Hamiltonian of a self-gravitating fluid system to the gravitational potential and others (kinetic and internal energies) and use differen
t time-steps for their integrations. The time integration is done in the way similar to that used in the mixed variable or multiple stepsize symplectic schemes. We performed three test calculations. One was the spherical collapse and the other was an explosion. We also performed a realistic test, in which the initial model was taken from a simulation of merging galaxies. In all test calculations, we found that the number of time-steps for gravitational interaction were reduced by nearly an order of magnitude when we adopted our integration method. In the case of the realistic test, in which the dark matter potential dominates the total system, the total calculation time was significantly reduced. Simulation results were almost the same with those of simulations with the ordinary individual time-step method. Our new method achieves good performance without sacrificing the accuracy of the time integration.
We show that the smoothed particle hydrodynamics (SPH) method, used with individual time-steps in the way described in the literature, cannot handle strong explosion problems correctly. In the individual time-step scheme, particles determine their ti
me-steps essentially from a local Courant condition. Thus they cannot respond to a strong shock, if the pre-shock timescale is too long compared to the shock timescale. This problem is not severe in SPH simulations of galaxy formation with a temperature cutoff in the cooling function at $10^4 {rm K}$, while it is very dangerous for simulations in which the multiphase nature of the interstellar medium under $10^4 {rm K}$ is taken into account. A solution for this problem is to introduce a time-step limiter which reduces the time-step of a particle if it is too long compared to the time-steps of its neighbor particles. Thus this kind of time-step constraint is essential for the correct treatment of explosions in high-resolution SPH simulations with individual time-steps.
We investigated the evolution of interacting disk galaxies using high-resolution $N$-body/SPH simulations, taking into account the multiphase nature of the interstellar medium (ISM). In our high-resolution simulations, a large-scale starburst occurre
d naturally at the collision interface between two gas disks at the first encounter, resulting in the formation of star clusters. This is consistent with observations of interacting galaxies. The probability distribution function (PDF) of gas density showed clear change during the galaxy-galaxy encounter. The compression of gas at the collision interface between the gas disks first appears as an excess at $n_{rm H} sim 10{rm cm^{-3}}$ in the PDF, and then the excess moves to higher densities ($n_{rm H} gtrsim 100{rm cm^{-3}}$) in a few times $10^7$ years where starburst takes place. After the starburst, the PDF goes back to the quasi-steady state. These results give a simple picture of starburst phenomena in galaxy-galaxy encounters.
We performed 3-dimensional N-body/SPH simulations to study how mass resolution and other model parameters such as the star formation efficiency parameter, C* and the threshold density, nth affect structures of the galactic gaseous/stellar disk in a s
tatic galactic potential. We employ 10^6 - 10^7 particles to resolve a cold and dense (T < 100 K & n_H > 100 cm^{-3}) phase. We found that structures of the ISM and the distribution of young stars are sensitive to the assumed nth. High-nth models with nth = 100 cm^{-3} yield clumpy multi-phase features in the ISM. Young stars are distributed in a thin disk of which half-mass scale height is 10 - 30 pc. In low-nth models with nth = 0.1 cm^{-3}, the stellar disk is found to be several times thicker, and the gas disk appears smoother than the high-nth models. A high-resolution simulation with high-nth is necessary to reproduce the complex structure of the gas disk. The global properties of the model galaxies in low-nth models, such as star formation histories, are similar to those in the high-nth models when we tune the value of C* so that they reproduce the observed relation between surface gas density and surface star formation rate density. We however emphasize that high-nth models automatically reproduce the relation, regardless of the values of C*. The ISM structure, phase distribution, and distributions of young star forming region are quite similar between two runs with values of C* which differ by a factor of 15. We also found that the timescale of the flow from n_H ~1 cm^{-3} to n_H > 100 cm^{-3} is about 5 times as long as the local dynamical time and is independent of the value of C*. The use of a high-nth criterion for star formation in high-resolution simulations makes numerical models fairy insensitive to the modelling of star formation. (Abridged)
