SPh

Sph, often written as SPH, stands for Smoothed Particle Hydrodynamics, a mesh-free, Lagrangian method for simulating continuum media such as fluids and gases. In SPH, the fluid is represented by a set of moving particles, each carrying properties like mass, position, velocity, density, and internal energy. Field quantities are estimated by smoothing over neighboring particles with a kernel function, enabling the calculation of densities and pressure forces without a fixed grid. The method was developed independently in the late 1970s by Lucy and by Gingold and Monaghan, and has since become widely used in astrophysics, engineering, and computer graphics.

In SPH, the density at a particle is computed by summing the masses of nearby particles weighted by the smoothing kernel. The equations of motion and energy follow from the fluid equations, applied in particle form; pressures are determined from an equation of state. The smoothing length, which sets the influence radius, can vary in space and time to maintain a roughly constant number of neighbors. Artificial viscosity terms are often included to capture shocks and prevent unphysical interpenetration, though modern formulations seek to reduce unnecessary dissipation.

Applications include modeling star and galaxy formation, accretion and protoplanetary disks, planetary formation, hydrodynamic flows with free surfaces, and visual effects in computer graphics. Advantages of SPH include natural handling of free surfaces and large deformations, straightforward handling of complex geometries, and adaptive resolution. Limitations involve lower accuracy for some flows, challenges with sharp discontinuities and contact interfaces, boundary treatment, and the need for careful tuning of viscosity and kernel choices. Variants and improvements exist, such as Godunov-SPH, kernel corrections, and entropy-based formulations, with ongoing development and multiple open-source implementations.