AN IMPROVED CONVERGENCE RESULT FOR THE SMOOTHED PARTICLE HYDRODYNAMICS METHOD

The smoothed particle hydrodynamics (SPH) method is a popular, kernel-based discretization method for fluid-flow problems. Despite its frequent use, mathematical understanding is still limited. In [T. Franz and H. Wendland, SIAM J. Math. Anal., 50 (2018), pp. 4752--4784] we proved convergence for a specific flow problem under appropriate conditions on the underlying kernel. The kernel has to satisfy so-called moment and approximation conditions. We also showed that the generalized Wendland kernels satisfy these conditions in odd space dimensions. In this paper, we will significantly improve the above results in the following ways. We will show that the results also hold in even space dimensions. We will show that the generalized Wendland kernels satisfy the approximation condition of any order, which means that for these kernels we can eliminate the dependence of the convergence rate on the approximation condition. We will show that the standard Wendland kernels, though they perform numerically similarly, do not satisfy the approximation condition.

Automated summary

The smoothed particle hydrodynamics (SPH) method is a popular discretization method for fluid-flow problems, but mathematical understanding is limited. This study improves previous results and extends the findings to even space dimensions, highlighting differences between different types of kernel functions.