A stable SPH discretization of the elliptic operator with heterogeneous coefficients

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Abstract

Smoothed particle hydrodynamics (SPH) has been extensively used to model high and low Reynolds number flows, free surface flows and collapse of dams, study pore-scale flow and dispersion, elasticity, and thermal problems. In different applications, it is required to have a stable and accurate discretization of the elliptic operator with homogeneous and heterogeneous coefficients. In this paper, the stability and approximation analysis of different SPH discretization schemes (traditional and new) of the diagonal elliptic operator for homogeneous and heterogeneous media are presented. The optimum and new discretization scheme of specific shape satisfying thetwo-point flux approximation nature is also proposed. This scheme enhances the Laplace approximation (Brookshaw’s scheme (1985) and Schwaiger’s scheme (2008)) used in the SPH community for thermal, viscous, and pressure projection problems with an isotropic elliptic operator. The numerical results are illustrated by numerical examples, where the comparison between different versions of the meshless discretization methods are presented.
Original languageEnglish
Article number112745
Number of pages32
JournalJournal of Computational and Applied Mathematics
Volume374
DOIs
Publication statusPublished - 2020

Keywords

  • Diffusive flow
  • Discrete maximum principle
  • Laplace operator
  • Meshless method
  • Monotone scheme
  • Stability and approximation

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