Towards an adaptive treecode for N-body problems

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Department of Mathematical Sciences


N-body problems are notoriously expensive to compute. For N bodies, evaluating a sum directly scales like O(N2). A treecode approximation to the N-body problem is highly desirable because for a given level of accuracy, the computation scales instead like O(NlogN). 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 Gϵ,n(x) and its derivatives.

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Journal of Scientific Computing