The Effect of Global Smoothness on the Accuracy of Treecodes (Communications in Computational Physics, 2023)
Treecode algorithms are widely used in evaluation of $N$-body pairwise interactions in $\mathcal{O}(N)$ or $\mathcal{O}(NlogN)$ operations. While they can provide high accuracyapproximations, a criticism leveled at the methods is that they lack global smoothness. In this work, we study the effect of smoothness on the accuracy of treecodesby comparing three tricubic interpolation based treecodes with differing smoothnessproperties: a global $\mathcal{C}^1$ continuous tricubic, and two new tricubic interpolants, one thatis globally $\mathcal{C}^0$ continuous and one that is discontinuous. We present numerical resultswhich show that higher smoothness leads to higher accuracy for properties dependenton the derivatives of the kernel, nevertheless the global $\mathcal{C}^0$ continuous and discontinuous treecodes are competitive with the $\mathcal{C}^1$ continuous treecode. One advantageof the discontinuous treecode over the $\mathcal{C}^1$ continuous is that, in addition to functionevaluations, the discontinuous treecode only requires evaluations of the first derivatives of the kernel while the $\mathcal{C}^1$ continuous treecode requires evaluations up to thirdorder derivatives. When the first derivatives are computed using finite differences, thediscontinuous version can be viewed as kernel independent and of utility for a widerarray of kernels with minimal effort.
Keywords: Fast summation; treecode; tricubic interpolation.