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Astron. Astrophys. 325, 857-865 (1997)

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2. The SPH methodology

In SPH one models a number of particles that carry the physical quantities, where the particles' distribution in space describes the density distribution. To simulate a fluid each particle's mass is smoothed over a radius r. Following for example Hernquist & Katz (1989), such a smoothed quantity at [FORMULA] be written for N particles as

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. The smoothing kernel, w, has the property

[EQUATION]

The kernel used here is the spline kernel from Monaghan and Lattanzio (1985):

[EQUATION]

This is a smooth kernel with compact support over the radius [FORMULA] around the particle. The smoothing length h is varying in time and updated every iteration to keep the interactions with other particles to a specific number. They are called the particle's neighbours, and the number of neighbours for each particle in the tests in this paper is 64.

In the SPH formulation there are different forms of the discretization of the Navier Stokes equations. Here the expressions of Hernquist & Katz (1989) are used, i.e.

[EQUATION]

where [FORMULA] and [FORMULA]. The continuity equation is automatically satisfied due to the Lagrangian formulation, and the density calculated from Eq. (1) becomes

[EQUATION]

The standard artificial viscosity term, [FORMULA], is defined

[EQUATION]

where [FORMULA]. The first and second term in the expression of [FORMULA] in Eq. (6) represents the bulk and the von Neumann Richtmeyer artificial viscosity respectively. The constant [FORMULA] is a fudge parameter to prevent the artificial viscosity to become too large. The artificial viscosity is used only when [FORMULA], that is when two particles are approaching each other. To close the system, the pressure is defined as [FORMULA], where u is the thermal energy density and [FORMULA] the adiabatic index. The particles' quantities are updated using a standard leapfrog integrator with the time step

[EQUATION]

where [FORMULA] is the Courant factor to stabilize the integration, and [FORMULA] the maximum µ from the interactions with the other particles. Since all particles are integrated with the same time step, the smallest time step from all particles is used in the integration. In the leapfrog integrator the velocity and internal energy density are integrated at half time steps, [FORMULA], while the position is integrated at whole time steps, [FORMULA], as

[EQUATION]

The viscous acceleration terms in Eq. (4) scales with [FORMULA] as

[EQUATION]

The artificial viscosity can therefore lead to undesirable effects, because the velocity differences are smoothed on a time scale of roughly [FORMULA]. The velocities in the model will therefore be smoothed out unless it expands or if there is some driving mechanism such as gravitation. To conserve the energy the particles are heated, which may be unphysical. The heated gas may reach an equilibrium state earlier than expected. A way of preventing interparticle penetration without unnecessary heating of the gas could therefore be useful, and this implies that there is a need to restrict the artificial viscosity to the shocks as much as possible.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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