Top : optimized kernel functions for the Gresho-Chan vortex (Sect. 3.4.2), for both GDSPH and ISPH together with the popular Wendland kernels (C₂, C₄). Middle : corresponding derivative of these kernel function. Bottom : fourier transform of these kernels, scaled by k/10. The optimized kernel exhibit a clear trend in it’s Fourier transform, where the kernels optimized for higher N_smooth emphasizes lower power beyond k > 3π. The general form of the optimized kernel is quite similar to the Wendland C₂ kernel. In general GDSPH produces lower power than ISPH in the high wave number regime for a given N_smooth. The O_I,32 kernel is the only optimized kernel that exhibits a negative Fourier transform at low wavenumbers. Among the other optimized kernels, only O₂₅₆ has a negative Fourier transform, but this occurs at higher wavenumbers and remains stable to pairing instability up to at least N_smooth < 500.