9 Additional Information9.4 Basics of HRSC methods

9.5 Newtonian SPH equations 

Following Monaghan [122Jump To The Next Citation Point In The Article] the SPH equation of motion for a particle a with mass m and velocity tex2html_wrap_inline6723 is given by


where the summation is over all particles other than particle a, p is the pressure, tex2html_wrap_inline5637 is the density, and d / dt denotes the Lagrangian time derivative. tex2html_wrap_inline6097 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., [77, 30]). tex2html_wrap_inline6103 is the interpolating kernel, and tex2html_wrap_inline6101 denotes the gradient of the kernel taken with respect to the coordinates of particle a .

The kernel is a function of tex2html_wrap_inline6741 (and of the SPH smoothing length tex2html_wrap_inline6743), i.e., its gradient is given by


where tex2html_wrap_inline6745 is a scalar function which is symmetric in a and b, and tex2html_wrap_inline6751 is a shorthand for tex2html_wrap_inline6753 . Hence, the forces between particles are along the line of centers.

Various types of spherically symmetric kernels have been suggested over the years [120, 12]. Among those the spline kernel of Monaghan & Lattanzio [123], 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 tex2html_wrap_inline6743, and has compact support.

In the Newtonian case tex2html_wrap_inline6097 is given by [122Jump To The Next Citation Point In The Article]


provided tex2html_wrap_inline6759, and tex2html_wrap_inline6127 otherwise. Here tex2html_wrap_inline6763, tex2html_wrap_inline6765 is the average sound speed, tex2html_wrap_inline6767, and tex2html_wrap_inline6769 is a parameter.

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 tex2html_wrap_inline5691 (see, e.g., [121]). 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 tex2html_wrap_inline6773, which reads [122Jump To The Next Citation Point In The Article]


where tex2html_wrap_inline6099 is the artificial energy dissipation term derived by Monaghan [122]. 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 [169Jump To The Next Citation Point In The Article, 172]. Steinmetz & Müller [169] 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.

9 Additional Information9.4 Basics of HRSC methods

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de