Localized kernel gradient correction for SPH simulations of water wave propagation

TL;DR

This paper tackles the high numerical dissipation of SPH in low-viscosity water-wave simulations by introducing a localized, weighted symmetric kernel gradient correction (CCSPH) that is applied only to a surface-proximate subset of particles. Guided by linear wave theory, the corrected subset is defined by depth using a parameter $\chi$, ensuring most kinetic energy resides near the surface is captured while reducing overhead; a parameter-free weighting based on kernel support stabilizes the gradient correction. Across a standing wave and a progressive wave train, the approach preserves wave period and energy far better than basic SPH, with damping characterized by $\beta$ matching theoretical expectations, and achieves meaningful computational savings (e.g., ~25% for a standing wave and ~10% for a long-wave tank) when the correction is restricted to the near-surface region. The results indicate that localized higher-order treatment can enable long-term, large-scale SPH simulations of deep-water waves with practical efficiency gains, and the method is generic enough to extend beyond linear wave scenarios.

Abstract

Basic Smoothed Particle Hydrodynamics (SPH) models exhibit excessive, numerical dissipation in the simulation of water wave propagation. This can be remedied using higher-order approaches such as kernel gradient correction, which introduce additional computational effort. The present work demonstrates, that the higher-order scheme is only required in a limited part of the water wave in order to obtain satisfying results. The criterion for distinguishing particles in need of special treatment from those that do not is motivated by water wave mechanics. Especially for deep water waves, the approach potentially spares large amounts of computational effort. The present paper also proposes a remedy for issues of the kernel gradient correction occurring at the free surface. Satisfying results for the proposed approach are shown for a standing wave in a basin and a progressive wave train in a long wave tank.

Figures (7)

• Figure 1: A propagating wave.
• Figure 2: Standing wave particle distribution computed by regularized CCSPH and $\chi=1/2$ at time $t=0.2$. For the sake of illustration, a large interparticle spacing of $\Delta p=1/64$ has been chosen. The left half of the domain displays the pressure field of the fluid particles, whereas the right half of the domain exhibits the different subsets $\mathcal{I}$ and $\mathcal{C}$ that result from the pressure criterion.
• Figure 3: Results for the standing wave computed by three different SPH formulations and $h/\Delta p=1.3$.
• Figure 4: Damping ratio over $\chi$ with fixed $\Delta p=1/256$.
• Figure 5: Damping ratio over millions of normalized iterations per particle per second for a fixed $\Delta p=1/256$.