A Stable SPH Discretization of the Elliptic Operator with Heterogeneous Coefficients

TL;DR

This paper develops a new stable SPH discretization scheme for the elliptic operator with heterogeneous coefficients, improving accuracy and stability for various physical simulations.

Contribution

It introduces an optimal and novel SPH discretization scheme that enhances existing Laplace approximations for heterogeneous media.

Findings

The new scheme improves stability over traditional methods.

Numerical examples demonstrate enhanced accuracy.

Comparison shows superiority of the proposed discretization.

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 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.