Unique solvability and stability analysis for incompressible smoothed particle hydrodynamics method

TL;DR

This paper provides a rigorous mathematical analysis of the unique solvability and stability of implicit and semi-implicit incompressible smoothed particle hydrodynamics (ISPH) schemes, introducing key conditions for stability and developing modified schemes.

Contribution

It introduces the first mathematical proof of solvability and stability for ISPH schemes, establishing key conditions and proposing modified schemes for automatic compliance.

Findings

Unique solvability in 2D and 3D under connectivity condition.

Stability of implicit scheme in 2D with connectivity and regularity.

Semi-implicit scheme stability with added time step condition.

Abstract

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.