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.

研究了求解不可压缩的Navier-Stokes方程的一种新的概率方案，其中我们近似了一个广义的非线性Feyman-Kac公式。速度场被解释为由布朗运动驱动的前向后向随机微分方程（FBSDE）所控制的随机过程的平均值。继Delbaen，Qiu和Tang在2015年提出的一种方法之后，压力项是通过计算与额外BSDE相关的积分函数的期望值的泊松问题从速度中获得的。遵循Delarue和Menozzi在2006年提出的算法，通过使用基于Euler型方案的前向后算法对本地时间积分和布朗运动增量进行量化来对FBSDE分量进行数值求解。关于不可压缩流体的Navier-Stokes方程的空间周期解析解报道了数值结果。我们在二维泰勒-格林涡旋和三维Beltrami流上说明了所提出的算法。