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By Garrett Birkhoff

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Let us now come to the precise definition of the method we plan to analyze. From now on, we make constant use of the notations and results given in Appendix A. Let Q =]0, 1[2, r > 1, and let coo be a periodic function in Q. Let f be a C°° cutoff function of order r. We define the periodic convolution of a periodic function (or distribution) / defined on Q with a function g defined in R2 by g *p f = g * / , where / denotes the periodic extension of / in R 2. Then, if £ is a cutoff function, we set f£(x) = e~ 2 £(x/£), a)e0 = ££ *, o)0, K£ = K * &.

6: the case of a bounded flow in a simply connected domain and the case of flows in periodic boxes. In the first case, we assume that the normal component of the velocity is zero, the so-called no-through-flow boundary condition. In terms of the stream function \j/ such that u = V x f, this means that the tangential derivative of \jr vanishes at the boundary. Since the boundary has a single connected component, \j/ is a constant, which can be set to zero as ty is obviously determined up to an additive constant, at the boundary.

For periodic or unbounded geometries, it is thus infinite-order accurate in the sense that the distance in some appropriate distribution space between the exact vorticity and its particle approximation can be bounded by Chm for all m, provided the vorticity has derivatives of an order up to m bounded. For rectangular domains without periodicity assumptions, the midpoint rule is only second order, but higher-order initializations can be obtained if initial particle locations coincide with quadrature points associated with Gauss-type quadrature formulas.

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