# On the efficiency of fast algorithms in 2D vortex element method

## Main Article Content

## Abstract

The analogue of the Barnes --- Hut algorithm is considered as one of the most efficient ways to acceleration of the velocities computation in vortex element method. When calculating the convective velocities this algorithm makes it possible to take into account the influence of vortex elements, which are located far enough from each other, approximately. The tree-based algorithm is developed for the calculation of diffusive velocities. The estimations of computational complexity are obtained for the algorithms for convective and diffusive velocities calculation. Also estimations for the errors of the vortex elements velocities computation are constructed, which depend on the algorithm parameters. These estimates allow in practice to choice the optimal parameters of the whole algorithm and achieve the maximum possible acceleration of the computations for the given maximum error level.

## Article Details

How to Cite

KUZMINA, Kseniia S.; MARCHEVSKY, Ilia K..
On the efficiency of fast algorithms in 2D vortex element method.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 93-102, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/398>. Date accessed: 22 sep. 2017.
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## References

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[13] K. S. Kuzmina, I. K. Marchevsky, Estimation of computational complexity of the fast numerical algorithm for calculating vortex influence in the vortex element method, Science & Education (electronic journal), 10 (2013), pp. 399–414 (URL: http://technomag.bmstu.ru/en/doc/604030.html)

[14] A. Grama, V. Sarin, and A. Sameh, Improving Error Bounds for Multipole-Based Treecodes, SIAM J. Sci. Comp., 21 (2000), pp. 1790–1803.

[15] J. K. Salmon, and M. S. Warren, Skeletons from the treecode closet, J. Comput. Phys., 111 (1994), pp. 136–155.

[2] P. Degond, and S. Mas-Gallic, The weighted particle method for convection-diffusion equations. Part I: The case of an isotropic viscosity, Math. Comp., 53 (1989), pp. 485–507.

[3] Y. Ogami, and T. Akamatsu, Viscous flow simulation using the discrete vortex model-the diffusion velocity method, Computers & Fluids, 19 (1991), pp. 433–441.

[4] G. Ya. Dynnikova, Lagrange method for Navier — Stokes equations solving, Doklady Akademii Nauk, 399 (2004), pp. 42–46.

[5] S. Guvernyuk, and G. Dynnikova, Modeling the flow past an oscillating airfoil by the method of viscous vortex domains, Fluid Dynamics, 42 (2007), pp. 1–11.

[6] I. K. Lifanov, and S. M. Belotserkovskii, Methods of Discrete Vortices. CRC Press, 1993.

[7] S. N. Kempka, M. W. Glass, J. S. Peery, and J. H. Strickland, Accuracy Considerations for Implementing Velocity Boundary Conditions in Vorticity Formulations. SANDIA Report SAND96-0583, 1996.

[8] K. S. Kuzmina, and I. K. Marchevsky The Modified Numerical Scheme for 2D FlowStructure Interaction Simulation Using Meshless Vortex Element Method, in Proc. IV Int. Conference on Particle-based Methods — Fundamentals and Applications (PARTICLES2015), Barcelona (2015), pp. 680–691.

[9] J. Barnes, and P. Hut, A hierarchical O(N logN ) force-calculation algorithm, Nature, 324 (1986), pp. 446–449.

[10] G. Ya. Dynnikova, Fast technique for solving the N -body problem in flow simulation by vortex methods, Computational Mathematics and Mathematical Physics, 49 (2009), pp. 1389– 1396.

[11] A. I. Gircha, Fast Algorithm for N -body Problem Solving with Regard to Numerical Method of Viscous Vortex Domains, Informatial technologies in Simulating and Control, 1 (2008), pp. 47–52 (in Russian).

[12] V. S. Moreva, On the Ways of Computations Acceleration when Solving 2D Aerodynamic Problems by Using Vortex Element Method, Heralds of the Bauman Moscow State University. Natural Sciences. Sp.Issue ‘Applied Mathematics’ (2011), pp. 83–95 (in Russian).

[13] K. S. Kuzmina, I. K. Marchevsky, Estimation of computational complexity of the fast numerical algorithm for calculating vortex influence in the vortex element method, Science & Education (electronic journal), 10 (2013), pp. 399–414 (URL: http://technomag.bmstu.ru/en/doc/604030.html)

[14] A. Grama, V. Sarin, and A. Sameh, Improving Error Bounds for Multipole-Based Treecodes, SIAM J. Sci. Comp., 21 (2000), pp. 1790–1803.

[15] J. K. Salmon, and M. S. Warren, Skeletons from the treecode closet, J. Comput. Phys., 111 (1994), pp. 136–155.