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

Abstract

A novel probabilistic scheme for solving the incompressible Navier-Stokes equations is studied, 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) driven by Brownian motion. Following an approach by Delbaen, Qiu and Tang introduced in 2015, the pressure term is obtained from the velocity by solving a Poisson problem as computing the expectation of an integral functional associated to an extra BSDE. The FBSDEs components are numerically solved by using 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.

Stochastic Approaches to Deterministic Fluid Dynamics: A Selective Review

“…We have formulated the problem in [28] first, then solved it in [29,30] for some specific function spaces and in two and three dimensions, respectively. The characterization in terms of forward-backward stochastic differential equations has also the advantage that it may allow to implement numerical methods, known for such systems (c.f for example, [31] and the recent work [32]).…”