Towards an Adaptive Treecode for N-body Problems

N-body problems are notoriously expensive to compute. For N bodies, evaluating a sum directly scales like $${\mathcal {O}}(N^2)$$. A treecode approximation to the N-body problem is highly desirable because for a given level of accuracy, the computation scales instead like $${\mathcal {O}}(N\log {N})$$. A main component of the treecode approximation, is computing the Taylor coefficients and moments of a cluster–particle approximation. For the two-parameter family of regularized kernels previously introduced (Ong et al. in J Sci Comput 71(3):1212–1237, 2017. https://doi.org/10.1007/s10915-016-0336-0), computing the Taylor coefficients directly is algebraically messy and undesirable. This work derives a recurrence relationship and provides an algorithm for computing the Taylor coefficients of two-parameter family of regularized kernels. The treecode is implemented in Cartesian coordinates, and numerical results verify that the recurrence relationship facilitates computation of $$G^{\epsilon ,n}({\mathbf {x}})$$ and its derivatives.