Unique solvability and stability analysis for incompressible smoothed particle hydrodynamics method

Abstract

The incompressible smoothed particle hydrodynamics (ISPH) method 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.

Appendix A Mathematical tools

1.1 Cauchy–Schwarz inequality

Let $M\in \mathbb {N}$. For all $a_i, b_i\in \mathbb {R}~(i=1,2,\ldots ,M)$, the following, called the Cauchy–Schwarz inequality, holds:

$$\begin{aligned} \sum _{i=1}^Ma_i b_i \le \left(\sum _{i=1}^M a_i^2\right)^{1/2}\left(\sum _{i=1}^M b_i^2\right)^{1/2}. \end{aligned}$$

(105)

1.2 Grönwall’s inequality

Let $M\in \mathbb {N}$. Assume that $a_i, b_i>0~(i=0,1,\ldots ,M)$, $c>0$ satisfy the inequality

$$\begin{aligned} a_k \le a_0 + c + \sum _{j=0}^{k-1} a_j b_j, \qquad k=1,2,\ldots ,M. \end{aligned}$$

(106)

Then, the following, called Grönwall’s inequality, holds:

$$\begin{aligned}&a_k \le (a_0+c)\prod _{j=0}^{k-1} (1+b_j) \le (a_0+c) \exp \left(\sum _{j=0}^{k-1} b_j\right), \nonumber \\&\quad k=1,2,\ldots ,M. \end{aligned}$$

(107)