Equalizing resolution in smoothed-particle hydrodynamics calculations using self-adaptive sinc kernels

The smoothed-particle hydrodynamics (SPH) technique is a numerical method for solving gas-dynamical problems. It has been applied to simulate the evolution of a wide variety of astrophysical systems. The method has a second-order accuracy, with a resolution that is usually much higher in the compressed regions than in the diluted zones of the fluid. In this work, we propose and check a scheme to balance and equalize the resolution of SPH between high- and low-density regions. This method relies on the versatility of a family of interpolators called Sinc kernels, which allows increasing the interpolation quality by varying only a single parameter (the exponent of the Sinc function). The scheme is checked and validated through a number of numerical tests, from standard one-dimensional Riemann problems in shock tubes, to multidimensional simulations of explosions, hydrodynamic instabilities and the collapse of a sun-like polytrope. The analysis of the hydrodynamical simulations suggests that the scheme devised to equalizing accuracy improves the treatment of the post-shock regions and, in general, of the rarefacted zones of fluids while causing no harm to the growth of hydrodynamic instabilities. The method is robust and easy to implement with a low computational overload. It conserves mass, energy, and momentum and reduces to the standard SPH scheme in regions of the fluid that have smooth density gradients.