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.

中文翻译：

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面向N体问题的自适应树码

摘要

众所周知，N体问题的计算成本很高。对于N个实体，求和直接像\（{\ mathcal {O}}（N ^ 2）\）那样缩放。非常需要N体问题的树码近似值，因为对于给定的准确性水平，计算会像\（{\ mathcal {O}}（N \ log {N}）\）那样进行缩放。树码近似的主要组成部分是计算簇-粒子近似的泰勒系数和矩。对于先前介绍的正则化内核的两参数系列（Ong等人在J Sci Comput 71（3）：1212-1237，2017. https://doi.org/10.1007/s10915-016-0336-0），直接计算泰勒系数在代数上是混乱且不理想的。这项工作推导了递归关系，并提供了一种算法来计算正则化核的两参数族的泰勒系数。树代码是在笛卡尔坐标系中实现的，数值结果证明了递归关系有助于\（G ^ {\ epsilon，n}（{\ mathbf {x}}）\）及其派生类的计算。