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A Fast Adaptive Vortex Method in Three Dimensions

### Thomas Buttke

### Abstract

Vortex methods are methods for solving time-dependent
incompressible flow problems by discretizing the vorticity into vortex elements
and following these elements in time.
The main difficulty with vortex methods as originally formulated
is that the cost of the evaluation of the velocity field induced
by N vortices is O(N^2).
This is expensive, particularly in three dimensions where the number of
elements can increase rapidly in time due to vortex stretching.
In this paper, we present a fast vortex method for incompressible flow
in three dimensions. It is based on two ideas. The first is an extension
of Anderson's Method of Local Corrections (MLC) algorithm, a particle-particle
particle mesh method in two dimensions, to the case of three dimensional
problems. In this approach, the calculation of the velocity field
induced by a collection of vortices is split into two parts:
(i) finite difference velocity field calculation using a fast Poisson solver,
the results of which are used to
represent the velocity field induced by vortices far from the evaluation
point; and (ii) an N - body calculation to compute the velocity
field induced by nearby vortices. The second idea is the use of
adaptive mesh refinement in the finite difference calculation.
Calculations with a vortex ring in three dimensions show
the breakeven point between the MLC with AMR and the
direct method is at N approximately 3000; for N approximately 64,000,
MLC with AMR can be twelve times faster than the direct method.