Random vortex methods for the Navier--Stokes equations

Two random vortex methods of Runge--Kutta type are presented for solving the two-dimensional Navier--Stokes equations. We intesgitate the accuracy of these methods by considering the model problem of a rotating flow with initial vorticity concentrated uniformly on a disk of finite radius. Functionals of the numerical solution are computed by Monte Carlo estimates with efficient variance reduction, and the results are compared to those obtained from Euler's method. The numerical results show that both the methods produce errors smaller by one power of the time step size than Euler's method, one seemingly even better than the other. These Runge--Kutta methods are derivations of similar schemes proposed by us in an earlier time for solving stochastic differential equations with constant diffusion coefficients. copyright 1988 Academic Press, Inc