SPH for incompressible free-surface flows. Part I: Error analysis of the basic assumptions

We provide a quantitative error analysis of the fundamental assumptions of weak compressibility and kernel interpolation in SPH for incompressible free-surface flows. This provides a framework for understanding many of the existing SPH variations. Weak compressibility generates spurious high-frequency acoustic solutions. We quantify acoustic eigensolutions, demonstrating that these can dominate the dynamics (more so than the kinematics) in SPH. Kernel interpolation consistency is affected when particle distributions are non-uniform and near boundaries. We show that for non-uniform distributions obtained from the SPH governing equations, second order consistency is retained. Near the free surface, however, kernel interpolation implicitly imposes local spurious accelerations. These, in the presence of weak compressibility, are manifest in high-frequency acoustic dynamics of the order of gravity g throughout the domain. We identify instability mechanisms in the presence of density non-uniformities that result in unstable temporal growth of both depth decaying and depth oscillatory modes. The latter are also identified as spurious high-frequency SPH errors. We illustrate all of these findings in the simplest case of a hydrostatic problem. Finally, we discuss the performance of existing treatments to SPH in the context of the present findings.