The Effect of Global Smoothness on the Accuracy of Treecodes

Treecode algorithms are widely used in evaluation of N-body pairwise in-teractions in O(N) or O(NlogN) operations. While they can provide high accuracy approximations, a criticism leveled at the methods is that they lack global smooth-ness. In this work, we study the effect of smoothness on the accuracy of treecodes by comparing three tricubic interpolation based treecodes with differing smoothness properties: a global C1 continuous tricubic, and two new tricubic interpolants, one that is globally C0 continuous and one that is discontinuous. We present numerical results which show that higher smoothness leads to higher accuracy for properties dependent on the derivatives of the kernel, nevertheless the global C0 continuous and discon-tinuous treecodes are competitive with the C1 continuous treecode. One advantage of the discontinuous treecode over the C1 continuous is that, in addition to function evaluations, the discontinuous treecode only requires evaluations of the first deriva-tives of the kernel while the C1 continuous treecode requires evaluations up to third order derivatives. When the first derivatives are computed using finite differences, the discontinuous version can be viewed as kernel independent and of utility for a wider array of kernels with minimal effort.

Automated summary

This work examines the impact of smoothness on the accuracy of treecodes and compares three tricubic interpolation methods with different smoothness properties. The results show that higher smoothness leads to higher accuracy, but the globally C0 continuous and discontinuous treecodes are competitive with the C1 continuous treecode.