A renormalization method for the evaluation of lattice sums

Berman, C.L.; Greengard, Leslie

doi:10.1063/1.530726

Cited by 92 publications

(53 citation statements)

References 9 publications

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“…The other strategies are based on fast Fourier transform 21 or recursive summation on hierarchical cell structures. 22,23 The first method has been used primarily to extend the original FMM to periodic systems. In this study, we also concentrate on the first method, showing that the summation can be carried out with great precision at a very low cost if an appropriate numerical procedure is employed.…”

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confidence: 99%

Precise and efficient Ewald summation for periodic fast multipole method

Amisaki

2000

J. Comput. Chem.

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This article deals with the Ewald summation method for efficiently handling contributions from periodic images in the framework of the fast multipole method (FMM). Although there have already been several reports regarding this method, the given formulas have been slightly different, thus leading to possible confusion. In this study, the summation formula for arbitrary lattice vectors is derived independently to show that this formula is identical to that of the cubic cell of Figueirido et al. ( J Chem Phys 1997, 106, 9835 and J Chem Phys 107, 7002). The correctness of the formula is confirmed by numerical tests carefully designed for the verification. In addition, a precise numerical method is proposed for subtracting the contributions of neighbor cells, which is a necessary operation in periodic FMM. Furthermore, an optimal choice is given for the parameters that control the convergence behavior of the summation formula. It is shown by numerical tests for a 50,000-particle system that a method with double-precision numbers gives force accuracies of 7, 11, and 14 digits when the degrees of expansion are 8, 16, and 32, respectively. These results indicate not only the correctness of the summation formula but also the effectiveness of the careful subtraction method. It is also shown that, at most, only 0.1% of the total cost of the force evaluation can be attributed to the summation. As a result, using the choice for parameters, the contributions from distant images can be taken into account with great precision at extremely low computational costs.

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confidence: 99%

Precise and efficient Ewald summation for periodic fast multipole method

Amisaki

2000

J. Comput. Chem.

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“…It becomes prohibitively expensive even with N at the level of O(100). The fast multipole methods developed by Greengard and Rokhlin, 14 Berman and Greengard 15 can be used in principle to reduce the operating account to cN 2 . However, the constant c could be quite large in practice.…”

Section: Introductionmentioning

confidence: 99%

Singularity formation in three-dimensional vortex sheets

Hou

Zhang

2003

Physics of Fluids

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We study singularity formation of three-dimensional ͑3-D͒ vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contributions of the integral equation. By introducing an appropriate change of variables, we show that the leading order vortex sheet equation degenerates to a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This change of variables is guided by a careful analysis based on properties of certain singular integral operators, and is crucial in identifying the leading order singular behavior. Our result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. We also show that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Moreover, we introduce a model equation for 3-D vortex sheets. This model equation captures the leading order singularity structure of the full 3-D vortex sheet equation, and it can be computed efficiently using fast Fourier transform. This enables us to perform well-resolved calculations to study the generic type of 3-D vortex sheet singularities. We will provide detailed numerical results to support the analytic prediction, and to reveal the generic form of the vortex sheet singularity.

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“…While our discussion has focused on the free-space problem, it is a straightforward matter to impose periodic boundary conditions on the computational domain. The necessary modifications are described in [15] for two-dimensional problems and in [4] for the three-dimensional case.…”

Section: Generalizations and Conclusionmentioning

confidence: 99%