On the Numerical Solution of the Poisson Equation in the Smoothed Particle Hydrodynamics

Abstract

In the present work, we address some aspects related to the numerical solution of the Poisson equation within the framework of the Smoothed Particle Hydrodynamics (SPH). In this case, the Poisson equation is solved using nonlocal, kernel-based discretizations of the differential operators (i.e., gradient, divergence, and Laplacian operators) defined on scattered particle sets without topological connections. This paper clarifies two distinct sources of pressure artifacts that are often conflated in the SPH literature: (i) checkerboard-like spurious pressure modes caused by discrete operator incompatibility and pressure-velocity decoupling, and (ii) loss of accuracy due to particle disorder and kernel support truncation near boundaries. We propose diagnostics and procedures that separate these mechanisms and use representative benchmark tests to compare common SPH Poisson/Laplacian formulations in terms of robustness, convergence, and sensitivity to disorder and boundary deficiency. The resulting perspective reconciles conflicting claims and provides a theoretical guidance for practical applications.