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**A particle method to solve transport-diffusion equations.**
*(English)*
Zbl 0678.35077

Numer. Math. (to appear).

A deterministic vortex method to solve the Navier-Stokes equations is presented and analysed in this paper. The method is based on a viscous time splitting of the operator into a convection part and a diffusion part as studied by J. T. Beale and A. Majda. The convection part of the equation is the Euler system and is solved by a classical vortex method. The diffusion part is the heat equation which is exactly solved by means of the Green kernel. At each time step, the vortex particles evolve during the convection step. They are allocated a weight which represents the vorticity strength and evolve after the diffusion time step. The order of convergence of the method is proved to be determined by the accuracy of the splitting algorithm and the regularity of the exact solution.

Reviewer: S.Mas-Gallic

### MSC:

35Q30 | Navier-Stokes equations |

35C05 | Solutions to PDEs in closed form |

65Z05 | Applications to the sciences |

35B65 | Smoothness and regularity of solutions to PDEs |