Unique solvability and stability analysis for incompressible smoothed particle hydrodynamics method

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.

[1]  J. Morris,et al.  Modeling Low Reynolds Number Incompressible Flows Using SPH , 1997 .

[2]  S. Cummins,et al.  An SPH Projection Method , 1999 .

[3]  Bachir Ben Moussa,et al.  On the Convergence of SPH Method for Scalar Conservation Laws with Boundary Conditions , 2006 .

[4]  B. Ben Moussa,et al.  Convergence of SPH Method for Scalar Nonlinear Conservation Laws , 2000, SIAM J. Numer. Anal..

[5]  S. Shao,et al.  INCOMPRESSIBLE SPH METHOD FOR SIMULATING NEWTONIAN AND NON-NEWTONIAN FLOWS WITH A FREE SURFACE , 2003 .

[6]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[7]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[8]  Philip M. Gresho,et al.  On the theory of semi‐implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory , 1990 .

[9]  Yoshimi Sonoda,et al.  A Stabilized Incompressible SPH Method by Relaxing the Density Invariance Condition , 2012, J. Appl. Math..

[10]  P. Raviart.  An analysis of particle methods , 1985 .

[11]  Rui Xu,et al.  Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach , 2009, J. Comput. Phys..

[12]  井元 佑介,et al.  Error estimates of generalized particle methods for the Poisson and heat equations , 2016 .

[13]  Steven J. Lind,et al.  High-order Eulerian incompressible smoothed particle hydrodynamics with transition to Lagrangian free-surface motion , 2016, J. Comput. Phys..

[14]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[15]  S. J. Lind,et al.  Incompressible smoothed particle hydrodynamics for free-surface flows: A generalised diffusion-based algorithm for stability and validations for impulsive flows and propagating waves , 2012, J. Comput. Phys..