Smoothed-particle hydrodynamics

Smoothed Particle Hydrodynamics (SPH) is a computational method used for simulating fluid flows. It is an extremely powerful technique and has been used in many fields of research, including astrophysics, ballistics, vulcanology and tsunami.

Method

The SPH method works by dividing the fluid into discrete particles, each representing a "fluid element". These particles have a spatial distance (known as the "smoothing length", typically represented in equations by h), over which their properties are "smoothed". This means that any physical quantity of any particle can be obtained by summing the relevant properties of all the particles which lie within two smoothing lengths. For example, the temperature of particle A depends on the temperatures of all the particles within 2h.

The contributions of each particle to a property are weighted according to how close they are to the particle of which we wish to know the properties. Mathematically, this is governed by the kernel function (symbol W). Kernel functions commonly used include a Gaussian function, and the cubic spline. The latter function is exactly zero for particles further away than two smoothing lengths (unlike the Gaussian, where there is a small contribution any finite distance away). This has the advantage of saving computational effort by not including contributions from distant particles.

The equation for any quantity A of particle i $A_{i}$ is given by the equation

$A_{i}(r)=\sum _{j}m_{j}{\frac {A_{j}}{\rho _{j}}}W(r-r',h),$

where $m_{j}$ is the mass of particle j, $A_{j}$ is the value of the quantity A for particle j, $\rho _{j}$ is the density associated with particle j, and W is the kernel function mentioned above.

Similarly, the spatial derivative of a quantity can be obtained by a similar method, using integration by parts to shift the del ( $\nabla$ ) operator from the physical quantity to the kernel function,

$\nabla A_{i}(r)=\sum _{j}m_{j}{\frac {A_{j}}{\rho _{j}}}\nabla W(r-r',h).$

The size of the smoothing length can be fixed in both space and time, however this does not take advantage of the full power of SPH. By giving each particle its own smoothing length, and allowing it to vary with time, the resolution of the simulation can be made to automatically adapt itself depending on local conditions. For example, in a very dense region where many particles are close together, the smoothing length can be made short, giving a high spatial resolution. Conversely, in low-density regions, the particles are far apart, and the resolution is low, saving computer time for the regions of interest.

Combined with an equation of state and an integrator, this method can simulate hydrodynamic flows extremely well, and since it is a Lagrangian method, its resolution is not limited by grid-cell spacing, unlike Eulerian methods.

Uses in Astrophysics

The adaptive resolution of smoothed particle hydrodynamics, combined with its ability to simulate phenomena covering many orders of magnitude, make it ideal for computations in theoretical astrophysics.

Simulations of galaxy formation, star formation, stellar collisions, supernovae and meteor impacts are some of the wide variety of astrophysical and cosmological uses of this method.

Generally speaking, SPH is used solely to model hydrodynamics. Incorporating other astrophysical processes which may be important like radiative transfer and magnetic fields is a current area of research in the community, with some success.

External Links

[First large simulation of star formation using SPH]