  The incompressible smoothed particle hydrodynamics method (ISPH) is a
numerical method widely used for accurately and efficiently solving flow
problems with free surface effects. However, to date there has been little
mathematical investigation of properties such as stability or convergence for
this method. In this paper, unique solvability and stability are mathematically
analyzed for implicit and semi-implicit schemes in the ISPH method. Three key
conditions for unique solvability and stability are introduced: a connectivity
condition with respect to particle distribution and smoothing length, a
regularity condition for particle distribution, and a time step condition. The
unique solvability of both the implicit and semi-implicit schemes in two- and
three-dimensional spaces is established with the connectivity condition. The
stability of the implicit scheme in two-dimensional space is established with
the connectivity and regularity conditions. Moreover, with the addition of the
time step condition, the stability of the semi-implicit scheme in
two-dimensional space is established. As an application of these results,
modified schemes are developed by redefining discrete parameters to
automatically satisfy parts of these conditions.
