Convergence study of the Chorin-Marsden formula

Lung-an Ying

School of Mathematical Sciences
Xiamen University
Xiamen, China
School of Mathematical Sciences
Peking University
Beijing, China

Abstract: The vortex method is applied to simulating viscous flows numerically. The advantage of the vorticity-stream function formulation is that it provides the motion of vorticies directly and it is suitable to high Reynolds' number flow. The difficulty of the vorticity-stream function formulation is that there is no explicit expression for the vorticity boundary conditions. A "vorticity creation operator" is introduced in the Chorin-Marsden formula to replace the boundary conditions. In this talk we will present our study on the convergence problem of the Chorin-Marsden formula for bounded domains. We first consider the exact solution and introduce an integral equation with Vorterra type for the density of the vortex sheet, then we are able to prove the convergence of the approximate solutions. The order of convergence is estimated, which is belived to be optimal. A comparison is made for different kinds of approximate vortex sheets.