Optimized smoothing kernels for smoothed particle hydrodynamics

We present a set of new smoothing kernels for smoothed particle hydrodynamics (SPH) that improve the convergence of the method without any additional computational cost. These kernels are generated through a linear combination of other SPH kernels combined with an optimization strategy to minimize the error in the Gresho-Chan vortex test case. To facilitate the different choices in gradient operators for SPH in the literature, we performed this optimization for both geometric density average force SPH (GDSPH) and linear-corrected gradient SPH (ISPH). In addition to the Gresho-Chan vortex, we also performed simulations of the hydrostatic glass, Kelvin-Helmholtz instability, Sod shocktube case, and Evrard collapse as well as a subsonic blob test. At low neighbor numbers ($<128$), there is a significant improvement across the different tests, with the greatest impact shown for GDSPH. Apart from the popular Wendland kernels, we also explored other positive-definite kernels in this paper, which include the "missing" Wendland kernels, Wu kernels, and the Buhmann kernel. In addition, we also present a method for producing arbitrary non-biased initial conditions in SPH. This method uses the SPH momentum equation together with an artificial pressure combined with a global and local relaxation stage to minimize local and global errors.