Fast Vortex Methods

We present three fast adaptive vortex methods for the 2D Eu-ler equations. All obtain long-time accuracy at almost optimal cost by using four tools: adaptive quadrature, free-Lagrangian formulation, the fast multipole method and a nonstandard error analysis. Our error analysis halves the diierentiability required of the ow, suggests an eecient new balance of smoothing parameters , and combines naturally with fast summation schemes. Numerical experiments with our methods connrm our theoretical predictions and display excellent long-time accuracy. INTRODUCTION Vortex methods solve the 2D incompressible Euler equations in the vorticity formulation by discretizing the Biot-Savart law with the aid of the ow map. They have been extensively studied , widely generalized and applied to complex high-Reynolds-number ows: See (Gustafson and Sethian, 1991) for a survey. Vortex methods involve several components; velocity evaluation , vortex motion, diiusion, boundary conditions and re-gridding. In this paper, we improve the speed, accuracy and robustness of the velocity evaluation. We eliminate the ow map, improve the quadrature used for the Biot-Savart law, and analyze the error in velocity evaluation in a nonstandard way, requiring less diierentiability of the ow and obtaining eecient new parameter balances. We employ standard techniques for the vortex motion and consider inviscid free-space ow to eliminate diiusion and boundary conditions. Our approach combines naturally with regridding and fast multipole methods. Lagrangian vortex methods move the nodes of a xed quadra-ture rule with the computed uid velocity, preserving the weights of the rule by incompressibility. This procedure loses accuracy when the ow becomes disorganized, motivating many regrid-ding techniques. Even before the ow becomes disorganized, however, obtaining high-order accuracy with xed quadrature weights requires smoothing of the singular Biot-Savart kernel. Smoothing gives high-order accuracy for short times but slows