Boosting the accuracy of SPH techniques: Newtonian and special-relativistic tests

Abstract

We study the impact of different discretization choices on the accuracy of SPH and we explore them in a large number of Newtonian and special-relativistic benchmark tests. As a first improvement, we explore a gradient prescription that requires the (analytical) inversion of a small matrix. For a regular particle distribution this improves gradient accuracies by approximately ten orders of magnitude and the SPH formulations with this gradient outperform the standard approach in all benchmark tests. Second, we demonstrate that a simple change of the kernel function can substantially increase the accuracy of an SPH scheme. While the "standard" cubic spline kernel generally performs poorly, the best overall performance is found for a high-order Wendland kernel which allows for only very little velocity noise and enforces a very regular particle distribution, even in highly dynamical tests. Third, we explore new SPH volume elements that enhance the treatment of fluid instabilities and, last, but not least, we design new dissipation triggers. They switch on near shocks and in regions where the flow -without dissipation-starts to become noisy. The resulting new SPH formulation yields excellent results even in challenging tests where standard techniques fail completely.

Improved Performances in Subsonic Flows of an SPH Scheme With Gradients Estimated Using an Integral Approach

“…This makes the method much less noisy than in the standard formulation, with accuracy in estimated gradients being greatly improved (García-Senz et al 2012;Rosswog 2015).…”

“…After the paper of García-Senz et al (2012), the performance of the scheme was investigated in detail by Rosswog (2015); here we briefly describe the essential features of the method.…”

“…However, such adjustments reflect negatively on the density estimation ability of the kernels. By introducing a non-zero central derivative, the kernel profile becomes steeper and this in turn implies an overestimate of density when compared with the corresponding B-spline (V12, Rosswog 2015).…”

“…The accuracy of density estimation in SPH for different kernel families has been assessed by many authors (Dehnen & Aly 2012;V12;Rosswog 2015;Zhu et al 2015). Here we measure the mean SPH density of N = 128 3 particles using a glass-like particle distribution inside a cube of sidelength unity and total mass one.…”

“…The resulting tensor scheme has been tested in a variety of hydrodynamical test cases (García-Senz et al 2012;Rosswog 2015), showing significant improvements as compared with the standard formulation. A crucial feature of the method is that it retains the Lagrangian nature of SPH, unlike previous attempts aimed at improving gradient accuracy.…”