A stable SPH discretization of the elliptic operator with heterogeneous coefficients

• Published in: Journal of computational and applied mathematics Vol. 374; p. 112745
• Main Authors: Lukyanov, Alexander A., Vuik, Cornelis
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.08.2020
• Subjects: Diffusive flow; Discrete maximum principle; Laplace operator; Meshless method; Monotone scheme; Stability and approximation; Laplace operator; Stability and approximation; Discrete maximum principle; Monotone scheme; Diffusive flow; Meshless method
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2020.112745

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. •Stable and accurate discretization of the elliptic operator is proposed.•Elliptic operator can be with homogeneous and heterogeneous diagonal coefficients.•Stability and approximation analysis of SPH discretization schemes is presented.•New SPH discretization scheme of the elliptic operator is proposed.•New scheme is optimum in the class of a two-point volumetric flux approximation.•New scheme enhances approximations proposed by Brookshaw (1985) and Schwaiger (2008)•New scheme is in line with an peridynamic formulation proposed recently.•New scheme subject to specific conditions is monotone.•New scheme allows to apply upwinding strategy during the solution of nonlinear PDEs.•Numerical analysis is performed for different kernel gradients.•Comparison with analytical solutions for boundary value problems is presented.•New scheme performance is demonstrated by SPE10 benchmark for filtration problems.