J. Garrett Jernigan and David H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables, and explicit accuracy controls. Astrophysical Journal Supplement Series, 71:871--893, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....well. The simulation of galaxies under gravitational force laws is the domain that hierarchical methods have been used most widely in so far, and we use it as being representative of nonuniform classical domains. Several hierarchical methods have been proposed to solve classical N body problems [2, 18, 3, 15, 29, 7]. The most widely used and promising among these are the Barnes Hut method [3] and the Fast Multipole Method [15] Between them, these two methods also capture all the important characteristics of hierarchical methods for classical N body problems (see Section 4.1.2) By using these methods to ....

J.G. Jemigan and D.H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables and explicit accuracy controls. Astrophysics Journal Supplement, page 871, 1989.

....large N body simulations. For example, Hernquist, Hut and Makino [15] Barnes and Hut [5] and Sellwood [31] explicitly say this; Singh, Hennessy and Gupta [32] have a master error equation in which clearly discreteness noise is dominant; and Pfenniger and Friedl [27] Jernigan and Porter [17], and Barnes and Hut [3] all imply that using the largest possible N is a desirable characteristic. ffl force softening, i.e. replacing r 2 with (r 2 2 sof t ) in the denominator of the gravitational force computation for some small constant sof t , usually chosen to approximate the ....

J. Garrett Jernigan and David H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables, and explicit accuracy controls. Astrophysical Journal Supplement Series, 71:871--893, 1989.

....research done on comparing data structures for N body simulation. More attention has been paid to the asymptotic performance of the algorithms, and to the accuracy properties of particular codes. A series of papers introduced particle cluster algorithms in the early to mid eighties[App85, JP89, BH86] with the version developed by Barnes and Hut [BH86] receiving the most attention with respect to implementation. The fastest bounds for the force computation problem are for the Fast Multipole Method of Greengard and Roklin[Gre88, GR87] which is a cluster cluster algorithm. Although the ....

J. G. Jernigan and D. H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables and explicit accuracy controls. The Astrophysical Journal Supplement, 71:871--893, 1989.

....research done on comparing data structures for N body simulation. More attention has been paid to the asymptotic performance of the algorithms, and to the accuracy properties of particular codes. A series of papers introduced particle cluster algorithms in the early to mid eighties[App85, JP89, BH86] with the version developed by Barnes and Hut [BH86] receiving the most attention with respect to implementation. The fastest bounds for the force computation problem are for the Fast Multipole Method of Greengard and Rokhlin[Gre88, GR87, CK95, RT96] which is a cluster cluster algorithm. ....

J. G. Jernigan and D. H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables and explicit accuracy controls. The Astrophysical Journal Supplement, 71:871--893, 1989.

....is the Particle Cluster algorithm. Our goal is to prove that a Particle Cluster algorithm based upon Nearest Neighbor Trees is as good as the more commonly used Barnes Hut algorithm[BH86] The Nearest Neighbor algorithm was developed independently by Benz et al. BBCP90] and by Jernigan and Porter [JP89], and has been observed to run very well in practice [Mak90] Heuristic arguments have been given to explain the good performance, but up until now there has been no rigorous proofs that it is a fast algorithm. We begin with a discussion of N body simulation in Astrophysics and present the generic ....

....time. Improvements on this run time have come by computing an approximation of the forces instead of exact forces. The particle cluster algorithm computes the force on each particle by using a spatial hierarchy to cluster particles. The algorithm was independently developed by several researchers [App85, BH86, JP89], with the algorithm due to Barnes and Hut being the one that is currently most widely used. The run time for the particlecluster algorithm is generally considered to be O(n log n) The Greengard Rokhlin algorithm [GR87, Gre87] is a cluster cluster algorithm which achieves O(n) run time by ....

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J. G. Jernigan and D. H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables and explicit accuracy controls. The Astrophysical Journal Supplement, 71:871--893, 1989.

....[10] choose separating planes based upon the point set, where the subdivision does not necessarily create equal sized regions. The bottom up up approach aims at grouping together points in a way that reflects the geometry of the particles. Independently, Benz et al. 2] and Jernigan and Porter [11] gave schemes where close together points are combined to form clusters. Although these data structures are much less understood than the top down approaches, they appear to perform well in practice [12] Performance Although the performance of the particle cluster algorithms is generally ....

....Can the theory of geometric separators [23] be used to build better trees 7.2 Nearest Neighbor Trees An alternative to the top down approach is to build the tree bottom up by combining close together particles. This method was independently proposed by Benz et al. 2] and by Jernigan and Porter [11]. This method appears to be competitive with the top down approaches [12] We define a nearest neighbor tree to be a tree that is formed by repeatedly collapsing mutually nearest neighbors 6 until a single point is left. This naturally gives a binary tree. We give an algorithmic definition, ....

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J. G. Jernigan and D. H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables and explicit accuracy controls. The Astrophysical Journal Supplement, 71:871, 1989.

.... because the N used in simulations usually is much smaller than the N of the system being modelled; force softening, which eliminates the singularity at r = 0 and lessens spurious collisional effects introduced by finite N sampling [9] using fast but approximate force algorithms such as tree codes [6, 45] or particlemesh methods (e.g. 61] numerical ordinary differential equation (ODE) integration truncation error; and machine roundoff error. The goal of the study of reliability is to ascertain what effect these input or local errors have on the output or global error. Less stringently, we can ....

....force computation algorithms are used, they are generally the next most dominant source of error. Barnes and Hut [7] analyzed their tree code [6] for errors and found that it works well for collisionless systems, but their results were inconclusive about collisional systems. Jernigan and Porter [45] introduce a tree method that is completely different from the Barnes and Hut algorithm, and works very well for collisional systems. Barnes and Hut [7] found that errors in global conservation of energy were dominated by force errors, and concluded that a modest error tolerance in force of about ....

J. Garrett Jernigan and David H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables, and explicit accuracy controls. Astrophysical Journal Supplement Series, 71:871--893, 1989.

....interaction and Biot Savart law) requires that one consider the contribution of N Gamma 1 terms in the update of each body. Algorithms which use a hierarchical data structure and an approximate force law for aggregates of bodies were introduced independently by Appel[1] Jernigan and Porter[2] and Barnes and Hut[3] Subsequently, a number of other authors have expanded on the theme of multipole approximations and hierarchical data structures. 4, 5, 6, 7, 8] The computer programs described by these authors are known collectively as treecodes because the underlying data structure in ....

....defined to have exactly one body [3] but it is sometimes desirable to construct trees whose terminal cells contain several bodies [11] Treecodes may be broadly classified according to the most complicated type of interaction that is explicitly evaluated by the implementation. Body Cell treecodes [3, 7, 2] compute interactions between individual bodies and cells in a hierarchical tree. These interactions are essentially the evaluation of the far field of a multipole expansion. Typically, O(N log N) interactions must be evaluated to find the accelerations on N bodies. Conversely, Cell Cell treecodes ....

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J. G. Jernigan and D. H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables and explicit accuracy controls. Ap. J. Suppl., 71, 871, (1989).

.... They separated force contributions in a slow varying force due to distant stars (regular component) and a highly fluctuating force induced by immediate neighbors (irregular component) More recently, temporal extrapolations have been introduced in the multi pole expansion by Jernigan and Porter [12], McMillan and Aarseth [18] and Sundaram [24] The immediate options to exploit different time scales are either to extrapolate force at slow moving particles or extrapolate contribution by distant particles (extrapolation of multi pole moESAIM: Proceedings, Vol. 1, 1996, pp. 197 211 V.M. ....

Jernigan, J.G. and Porter, D.H., "A Tree Code with Logarithmic Reduction of the Force Terms, Hierarchical Regularization of all Variables, and Explicit Accuracy Controls," Astrophysical J. Suppl. Series, 71, pp. 871--893, 1989.

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J. Garrett Jernigan and David H. Porter. A tree code with logarithmic reduction of force terms, hierarchical regularization of all variables, and explicit accuracy controls. Astrophysical Journal Supplement Series, 71:871--893, 1989.

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