Mixing Sinc kernels to improve interpolations in smoothed particle hydrodynamics without pairing instability

TL;DR

This paper introduces a novel mixing of Sinc kernels for smoothed particle hydrodynamics that enhances interpolation accuracy, resists pairing instability, and performs well across varying neighbor counts, improving turbulence and vortex simulations.

Contribution

It develops a linear combination of low- and high-order Sinc kernels that improves interpolation quality and stability in SPH without increasing computational costs significantly.

Findings

Resistant to pairing instability across 60 to 400 neighbors.

Improves accuracy and convergence in vortex simulations.

Applicable to other kernel families like B-splines and Wendland.

Abstract

The smoothed particle hydrodynamic technique is strongly based on the proper choice of interpolation functions. This statement is particularly relevant for the study of subsonic fluxes and turbulence, where inherent small errors in the averaging procedures introduce excessive damping on the smallest scales. To mitigate these errors we can increase both the number of interpolating points and the order of the interpolating kernel function. However, this approach leads to a higher computational burden across all fluid regions. Ideally, the development of a single kernel function capable of effectively accommodating varying numbers of interpolating points in different fluid regions, providing good resolution and minimal errors would be highly desirable. In this work, we revisit and extend the main properties of a family of interpolators called $Sinc~kernels$ and compare them with the widely used family of Wendland kernels. We show that a linear combination of low- and high-order Sinc kernels generates good-quality interpolators, which are resistant to pairing instability while maintaining good sampling properties in a wide range of neighbor interpolating points, $60\le n_b\le 400$. We show that a particular case of this linear mix of Sincs produces a well-balanced and robust kernel that improves previous results in the Gresho-Chan vortex experiment even when the number of neighbors is not large, while yielding a good convergence rate. Although such a mixing technique is ideally suited for Sinc kernels owing to their excellent flexibility, it can be easily applied to other interpolating families such as the B-splines and Wendland kernels.