A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations

We study a probabilistic numerical scheme to solve the incompressible Navier-Stokes equations, in which we approximate a generalized nonlinear Feyman-Kac formula. The velocity field is interpreted as the mean value of a stochastic process ruled by Forward-Backward Stochastic Differential Equations (FBSDEs). Using a representation by Del-baen, Qiu and Tang introduced in 2015, the pressure term is obtained from the velocity by solving a Poisson problem as the expectation of an integral functional associated to an extra BSDE. The FBSDEs components are numerically solved by following a forward-backward algorithm based on Euler type schemes for the local time integration and the quantization of the increments of Brownian motion following the algorithm proposed by Delarue and Menozzi in 2006. Numerical results are reported on spatially periodic analytic solutions of the Navier-Stokes equations for incompressible fluids. We illustrate the proposed algorithm on a two dimensional Taylor-Green vortex and three dimensional Beltrami flows.