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Abstract

Abstract. Recently several different approaches have been developed for the simulation of three-dimensional incompressible fluid flows using vortex methods. Some versions use detailed tracking of vortex filament structures and often local curvatures of these filaments, while other methods require only crude information, such as the vortex blobs of the two-dimensional case. Can such "crude" algorithms accurately account for vortex stretching and converge? We answer this question affirmatively by constructing a new class of "crude" three-dimensional vortex methods and then proving that these methods are stable and convergent, and can even have arbitrarily high order accuracy without being more expensive than other "crude" versions of the vortex algorithm.1. Introduction. Vortex methods have provided an attractive and successful approach for the numerical simulation of incompressible fluid flow at high Reynolds number in two dimensions. They have several distinctive features: (1) the interactions of the computational vortices mimic the physical mechanisms in actual fluid flow; (2) vortex methods are automatically adaptive, since the vortex "blobs" concentrate in the regions of physical interest; and (3) there are no inherent errors which behave like the numerical viscosity of conventional Eulerian difference methods. Such diffusive errors can swamp the effects of physical viscosity in high Reynolds number flow similation.The earliest attempts to represent flows by a vortex method used point vortices, and thus permitted arbitrarily large local velocities (Rosenhead [20]). Chorin [3] and Kuwahara and Takami [24] introduced the idea of using a finite vortex core, rather than a point vortex, to stabilize the method. Subsequently there have been a large