Newtonian SPH equations

9.6 Newtonian SPH equations

Following Monaghan [200

], the SPH equation of motion for a particle a with mass $m$ and velocity $v$ is given by

where the summation is over all particles other than particle a, $p$ is the pressure, $ρ$ is the density, and $d∕dt$ denotes the Lagrangian time derivative. $Πab$ is the artificial viscosity tensor, which is required in SPH to handle shock waves. It poses a major obstacle in extending SPH to relativistic flows (see, e.g., [130 , 53 ]). $Wab$ is the interpolating kernel, and $∇aWab$ denotes the gradient of the kernel taken with respect to the coordinates of particle a.

The kernel is a function of $|ra − rb|$ (and of the SPH smoothing length $hSPH$ ), i.e., its gradient is given by

where $Fab$ is a scalar function which is symmetric in $a$ and $b$ , and $rab$ is a shorthand for $ra − rb$ . Hence, the forces between particles are along the line of centers.

Various types of spherically symmetric kernels have been suggested over the years [198 , 20 ]. Among those the spline kernel of Monaghan and Lattanzio [201 ], mostly used in current SPH codes, yields the best results. It reproduces constant densities exactly in 1D, if the particles are placed on a regular grid of spacing $hSPH$ , and it has compact support.

In the Newtonian case $Πab$ is given by [200 ]

( ( ) || hSPHvab-⋅ rab - hSPHvab-⋅ rab { − α ρ- |r |2 cab − 2 |r |2 for vab ⋅ rab < 0, Πab = || ab ab ab (103 ) ( 0 otherwise.

Here $vab = va − vb$ , $- cab = (ca + cb)∕2$ is the average sound speed, $-- ρab = (ρa + ρb)∕2$ , and $α ∼ 1.0$ is a parameter. Alternative forms of the artificial viscosity are discussed in [203 , 258 ].

Using the first law of thermodynamics and applying the SPH formalism, one can derive the thermal energy equation in terms of the specific internal energy $𝜀$ (see, e.g., [199 ]). However, when deriving dissipative terms for SPH guided by the terms arising from Riemann solutions, there are advantages to use an equation for the total specific energy $2 E ≡ v ∕2 + 𝜀$ , which reads [200 ]

where $Ωab$ is the artificial energy dissipation term derived by Monaghan [200 ]. For the relativistic case the explicit form of this term is given in Section 4.2.

In SPH calculations the density is usually obtained by summing up the individual particle masses, but a continuity equation may be solved instead, which is given by

The capabilities and limits of SPH have been explored, e.g., in [269 , 16 , 167 , 275 ]. Steinmetz and Müller [269 ] conclude that it is possible to handle even difficult hydrodynamic test problems involving interacting strong shocks with SPH, provided a sufficiently large number of particles is used in the simulations. SPH and finite volume methods are complementary methods to solve the hydrodynamic equations each having its own merits and defects.