## Minimalist introduction to SPH

In Smoothed Particle Hydrodynamics, the fluid is represented as a set
of N_{part} particles.
The position of these particles are evolved according to equations of
motion that incorporate gravitational and pressure forces. The particles
are also given thermal energies which evolve adiabatically. Artificial
viscosity terms are added to simulate shocks discontinuities.

Each particle is ascribed an interpolation kernel function W of spherical symmetry and variable
size h (``smoothing length'').
This allows the easy evaluation of fields and their derivatives. For
instance, the iterpolated value of (scalar) field A and its gradient gradA
at position r are
given by:

where m_{j}, rho_{j}, and A_{j} are the mass, density and A values for particle j. In fact, the proper estimate of
the density at the position of particle i is not but rho_{j} but

Hence particle i can be seen as
a cloud spread inside a sphere of radius 2h_{i}.
Only neighboring particles inside this sphere act on part. i with pressure forces. From the
point of view of gravitation, SPH is completely similar to N-body
approaches to which it borrows algorithms to compute gravitational
forces more efficiently than the N_{part}^{2}
direct summation, for instance a binary tree (Press
1986, Benz
et al. 1990).

When featured with a tree to compute gravitation, SPH is a grid-less
method which can cope with any asymmetrical three dimensional geometry.
It ignores void spaces completely, imposes no physical limits beyond
which matter cannot be tracked, does not come into trouble with large
dynamic range as long as variable smoothing lengths are implemented. In
situations with highly anisotropic compressions, like the tidal
interaction between a star and a massive black hole, SPH particles may,
however, loose contact in directions of lowest compression. To cure
this problem, SPH codes have been proposed that use ellipsoidal kernels
(Fulbright
et al. 95). The drawback of these schemes is that angular momentum
is not exactly conserved anymore so they can only be applied in special
cases where this will not endanger the physical significance of
results. SPH is better suited to highly dynamical problems than to
near-equilibrium configurations (Steinmetz
& Mueller 93).