A fast adaptive vortex method using local corrections

Vortex methods are particle methods for solving time-dependent incompressible fiow problems by discretizing the vorticity into vortex elements and following these elements in thne. The main difficulty with vorte..x methods as originally formulated is that the cost of the evaluation of the velocity field induced by _N' vortices is O( N'2). Tlus is expellsive. particularly in three dimensions where the number of eleillents can increa.se rapidly in time due to vortex stretching. Several lllethods have been developed for reducing this cost by exploiting the fact that the velocity induced by a. vortex elelnellt is harmonic; the method of local corrections (~ILC) is one such method. The 1-ILC is a particle-particle particle-mesh nlethod~ in which the calcula.tion of the velocity field induced by a collection of vortices is split into two parts: (i) a finite difference velocity field calculation using a fast Poisson solver on a unifornl grid superimposed on the vorticity field, and interpolation of tha,t velocity frolll the grid points to the vortices; (li) a local corrections step in which local interactions are computed directly. 'Ve present a fast adaptive vortex method which a.dds adaptive mesh refinement to the ~1LC.

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