the Navier–Stokes equations. It needs special consideration because it involves the Laplacian differential operator. Since the direct computation does not provide satisfactory results, several approaches to model the diffusion have been proposed. Artificial viscosity Introduced by Monaghan and Gingold the artificial viscosity was used to deal with high Mach number fluid flows. It reads Here, is controlling a volume viscosity while acts similar to the Neumann Richtmeyr artificial viscosity. The is defined by The artificial viscosity also has shown to improve the overall stability of general flow simulations. Therefore, it is applied to inviscid problems in the following form It is possible to not only stabilize inviscid simulations but also to model the physical viscosity by this approach. To do so is substituted in the equation above, where is the number of spartial dimensions of the model. This approach introduces the bulk viscosity . Morris For low Reynolds numbers the viscosity model by Morris was proposed. LoShao Additional physics Surface tension Heat transfer Turbulence Multiphase extensions Astrophysics Often in astrophysics, one wishes to model self-gravity in addition to pure hydrodynamics. The particle-based nature of SPH makes it ideal to combine with a particle-based gravity solver, for instance tree gravity code, particle mesh, or particle-particle particle-mesh. Solid mechanics and fluid-structure interaction (FSI) Total Lagrangian formulation for solid mechanics To discretize the governing equations of solid dynamics, a correction matrix is first introduced to reproducing rigid-body rotation as where stands for the gradient of the kernel function evaluated at the initial reference configuration. Note that subscripts and are used to denote solid particles, and smoothing length is identical to that in the discretization of fluid equations. Using the initial configuration as the reference, the solid density is directly evaluated as where is the Jacobian determinant of deformation tensor . We can now discretize the momentum equation in the following form where inter-particle averaged first Piola-Kirchhoff stress is defined as Also and correspond to the fluid pressure and viscous forces acting on the solid particle , respectively. Fluid-structure coupling In fluid-structure coupling, the surrounding solid structure is behaving as a moving boundary for fluid, and the no-slip boundary condition is imposed at the fluid-structure interface. The interaction forces and acting on a fluid particle , due to the presence of the neighboring solid particle , can be obtained as and Here, the imaginary pressure and velocity are defined by where denotes the surface