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Towards an Adaptive Treecode for N -body Problems
Journal of Scientific Computing ( IF 2.8 ) Pub Date : 2020-03-05 , DOI: 10.1007/s10915-020-01177-1
Benjamin W. Ong , Satyen Dhamankar

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:

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。

更新日期:2020-03-20
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