A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations. (English) Zbl 07506618
Summary: 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 F. Delbaen et al. [Stochastic Processes Appl. 125, No. 7, 2516–2561 (2015; Zbl 1323.35010)] 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 [F. Delarue and S. Menozzi, Ann. Appl. Probab. 16, No. 1, 140–184 (2006; Zbl 1097.65011)]. 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.
Keywords:
Navier-Stokes equations; Brownian motion; Feynman-Kac formula; forward-backward stochastic differential equations; incompressible fluids; numerical solutionReferences:
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