Abstract
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.
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Notes
The computational timing tests were performed on Superior, a high-performance computing infrastructure at Michigan Technological University. The research-grade C++ code was compiled without optimization using the GNU g++-4.4.6 compiler. The compute node that performed the computation had dual Intel Xeon E4-2680 2.50 GHz).
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Ong, B.W., Dhamankar, S. Towards an Adaptive Treecode for N-body Problems. J Sci Comput 82, 72 (2020). https://doi.org/10.1007/s10915-020-01177-1
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DOI: https://doi.org/10.1007/s10915-020-01177-1