Quadratic Convergence of Vortex Methods

We prove quadratic convergence for two-dimensional vortex methods with positive cutoffs. The result is established for flows with initial vorticity three times continuously differentiable and compact support. The proof is based on a refined version of a convergence result. Introduction. The purpose of this paper is to prove that vortex methods with positive cutoffs can converge quadratically if the cutoff length is proportional to the mesh length and the flow is sufficiently smooth. This has been observed computationally by Hald and Del Prête [13], Beale and Majda [6] and Perlman [19]. The vortex method is a numerical technique for approximating the flow of an incompressible, inviscid fluid. The flow is described by Euler's equations. The method for the two-dimensional case was introduced by Chorin (see [8]). Various three-dimensional methods have been suggested and studied by Chorin [9], Beale and Majda [4], Greengard [12], Anderson and Greengard [2], Leonard [16], Raviart [20] and Beale [3]. Recently, Chiu and Nicolaides [7] investigated a vortex method with nonuniform mesh and a higher-order quadrature formula. The convergence of the vortex method was first proved by Hald and Del Prête [13], but only for a short time interval. They assume that the initial vorticity is Holder continuous and their class of cutoff includes some that are positive and singular. Positive cutoffs were not included in the theory of Hald [14], but were covered in the study of Beale and Majda [5]. They proved higher-order convergence for smooth flows and cutoffs that satisfy the so-called moment conditions and almost quadratic convergence for positive cutoffs. Our class of cutoffs cannot be compared with Beale and Majda's [5]. We assume more smoothness at the origin but allow a slow decay at infinity. In this paper we assume that the vorticity is three times continuously differentiable and prove quadratic convergence for our class of cutoffs. If the vorticity is two times continuously differentiable, we only obtain almost quadratic convergence. If the cutoff is positive, our result is better than the result of Beale and Majda [5]. On the other hand, Beale and Majda's theory gives higher rate of convergence for higher-order cutoffs. Our proof breaks down if the flow is not smooth. For such a flow Hald [15] has proved superlinear convergence for a large class of cutoffs. It has been customary in previous papers [13], [14], [5], [1], [2], [15] to assume that the mesh length tends to zero faster than the cutoff length. It has been even Received September 29, 1986; revised November 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 65M15, 76C05. * Current address: Istituto di Matemática, Università di Genova, 16132 Genova, Italy ©1989 American Mathematical Society 0025-5718/89 $1.00 + $.25 per page