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

https://doi.org/10.1016/j.jcp.2020.109689Get rights and content

Highlights

  • Computer simulation of the unsteady Navier-Stokes equations on two and three dimensions.

  • Numerical solution of a novel FBSDEs system associated to the Navier-Stokes equations.

  • Numerical performance of the probabilistic algorithm on incompressible fluids.

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.

Introduction

The Navier-Stokes equations system allows us to model the movement of fluids. It was introduced by C.-L. Navier in 1822 [1] and G. G. Stokes in 1849 [2] by incorporating a pressure term and the fluid viscosity to the Euler equations due to L. Euler in 1757 [3]. Nowadays, technological and scientific computing developments provide a suitable way for the simulation of mathematical models, and in turn research and analysis of efficient numerical methodologies to approximate exact solutions. With this in mind, we study the numerical simulation of the Navier-Stokes equations for incompressible fluids in Rd (for space dimension d{2,3}){ut+(u)u=νup+f;0<tT,u=0,u(0)=g, where T>0 is a fixed time, ν>0 is the kinematic viscosity, f is the external force field and g is a given initial divergence-free vector field. The system of equations (1) describes the motion of an incompressible fluid by means of unknown fields of velocity u(t,x)Rd and pressure p(t,x)R defined for each time t[0,T] and position xRd. It involves a nonlinear convective term (u)u, a diffusion term νu and the incompressibility condition u=0. Although being introduced in the nineteenth century, there are still open problems concerning the Navier-Stokes equations. Nonetheless it is usual to find works that deal with the fluid dynamics using the Navier-Stokes model (ocean modeling, atmosphere of Earth, gas dynamics, dynamic of storms, etc.) [4], [5], [6], [7]. There exists a huge literature on computational fluid dynamics, and probabilistic numerical methods for nonlinear models have gained attention during the last decades.

The Itô stochastic differential equations (SDEs) introduced during the 1940s by K. Itô's seminal works [8], [9] are connected with partial differential equations (PDEs) by means of the celebrated Feynman-Kac formula due to R. Feynman [10] and M. Kac [11]. More recently, the theory of backward stochastic differential equations (BSDEs), initiated by É. Pardoux and S. Peng in [12], permits to obtain probabilistic representations for the solutions of a broad class of PDEs by means of systems of forward-backward SDEs (FBSDEs) through a nonlinear Feynman-Kac formula. Hence, the numerical solution of FBSDEs provides a stochastic algorithm to approximate solutions of nonlinear PDEs. Among the theory of SDEs, FBSDEs associated to the unsteady Navier-Stokes equations is a novel approach.

Let us assume that the solution u of (1) is known on a time interval [0,T], which means that the pressure term p is also known (the force field f is considered as given). We define by X the d-dimensional diffusion process associated to the differential operator u(u)u+νu. With Ys:=u(Ts,Xs) and Zs:=Du(Ts,Xs), the process (X,Y,Z) is solution to the FBSDEs{Xs=x+tsYrdr+ts2νdWr,Ys=g(XT)+sT[p(Tr,Xr)f(Tr,Xr)]drsT2νZrdWr, where Ws is a d-dimensional standard Brownian motion. The numerical solution of the incompressible Navier-Stokes equations (1) involves various difficulties due to the presence of a nonlinear term, the incompressibility condition as well as the computation of the pressure term. The simplified model obtained by removing the pressure term from the FBSDEs (2) is associated to the Burgers equation [13], which remains a nonlinear equation due to the term (u)u. Under appropriate conditions, the pressure term p is interpreted by solving a Poisson problem p=P(u) for a given functional of the solution.

Recently F. Delbaen, J. Qiu and S. Tang introduced in [14] a new class of coupled FBSDEs associated to the incompressible Navier-Stokes equations. Since their probabilistic approach involves an extra BSDE defined on an infinite time interval for the stochastic representation of the pressure term, Delbaen et al. deduced an approximated solution to the velocity field by truncating the infinite interval of the associated FBSDEs system. Hence they proposed a numerical simulation algorithm, that follows a methodology of F. Delarue and S. Menozzi [15], [16] for FBSDEs, to simulate the incompressible Navier-Stokes equations in the whole Cartesian space.

Specialized literature on probabilistic representations to the solution of PDEs is presented through Feynman-Kac formulae or well by means of branching processes. In general, the deterministic solutions of particular equations are obtained through the expectation of functionals of stochastic processes. During the last years some probabilistic approaches have been proposed to deal with the Navier-Stokes equations. The random vortex methods study the incompressible Navier-Stokes system in vorticity form [17], [18], [19]. Branching processes and stochastic cascades present another way of interpretation [20], [21], [22], [23], [24]. The stochastic Lagrangian paths provide additional representations [25], [26], [27]. Moreover, stochastic particle systems arise from the McKean-Vlasov equations in both two and three dimensional contexts [28], [29], [30]. Recently the systems of FBSDEs appear as a novel approach [31], [14]. The problem of relating boundary conditions on probabilistic representations presents additional challenges [32], [33], [34], [35].

The main motivation of this paper is to develop a suitable numerical framework formulated on the new probabilistic representation of Delbaen et al. for solving the incompressible Navier-Stokes equation (1) in two and three spatial dimensions. The main point is to test if such a method is practically feasible, while Monte Carlo methods are in general difficult to implement for such non-linear equations. Monte Carlo methods however offer the advantage of being simple to interpret and to set-up. By considering the numerical solution associated to the system of FBSDEs and their approximation by mean values, we study a backward regression based on the Euler-Maruyama scheme and the quantization method. The pressure gradient is recovered from the expected value of an integral functional of Brownian motion. To do this, we use the Riemann sum estimation and the quantization of the Brownian increments.

Finally, we test the performance of the novel stochastic algorithm to the simulation of spatially-periodic analytic solutions in the unsteady regime. In our simulations, the magnitude of Reynolds number ranges from 1 to 200. The numerical tests show that the proposed scheme is practically implementable. However, it suffers from long run-times which motivate the development of ad hoc variance reduction schemes. This article is only a preliminary step to the development of a Monte Carlo method for solving the incompressible Navier-Stokes equation.

Section 2 details the link between FBSDEs driven by Brownian motion and deterministic PDEs through the nonlinear Feynman-Kac formula. The incompressible Navier-Stokes equations are associated to the novel FBSDEs system. Section 3 presents the numerical methodology for solving FBSDEs. We detail the approximation of mean values and the numerical treatment of integral functionals of Brownian motion. Section 4 illustrates the forward-backward probabilistic algorithm by solving a two-dimensional Taylor-Green vortex and three-dimensional Beltrami flows as numerical examples. Finally Section 5 is devoted to concluding remarks.

We use the following notations and conventions:

  • We consider a filtered complete probability space (Ω,F,(Ft)t0,P).

  • For each q1, we setLFq(Ω;Rd):={ξ:ΩRd:ξ is F-measurable andEξq<}.

  • For each m,nZ+ and ARm×n, we denote by A its transpose matrix while Im represents the m×m identity matrix. The matrix ARn×n is said to be elliptic if there exists λ>0 such thatx,Axλx;xRn. The set of m×m real valued symmetric non-negative matrices isS+m:={QRm×m:Q=Q and x,Qx0;xRm}.

  • The symbol “≈” means “is numerically approximated by”.

  • The symbol O means the order of estimation. For example, h=O(δ2) means that h>0 is depending on δ>0 and less than Kδ2, with constant K independent of δ. From now on K>0 stands for constants independent on the discretization parameters.

  • The indicator function of a set A is 1A.

  • In general, we use the Einstein summation convention through this paper.

  • The space Ck,α(Rd,Rn), or simply as Ck,α, of k-continuously differentiable functions ϕ:RdRn with α Hölder continuity for each k-order partial derivatives equipped with the normϕCk,α:=supxRdϕ(x)+|β|=1ksupxRdDβϕ(x)+|β|=ksupx,yRd,xyDβϕ(x)Dβϕ(y)xyα. The multi-index notation is considered.

  • For T>0 and t[0,T) we denote by C([t,T];Rm×n) the space of continuous functions ψ with normψC([t,T];Rm×n):=sups[t,T]ψ(s). Similarly we have the space C([t,T];Ck,α) of continuous functions withψC([t,T];Ck,α):=sups[t,T]ψ(s)Ck,α.

  • Let LF2(Ω×[t,T];Rm×n) the space of predictable stochastic processes Ψ with values in Rm×n such thatE(tTΨs2ds)1/2<.

  • Let LF2(Ω;C([t,T];Rm×n)) be the set of all {Ft}t0-progressively measurable stochastic processes Ψ with continuous trajectories taking values in Rm×n such thatEsups[t,T]Ψs2<.

  • For mN and q[1,) we denote by Lq(R) and Hm,q(R) the usual R-valued Lebesgue and Sobolev spaces on Rd, respectively. When =d we just write Lq and Hm,q, or Hm when q=2.

  • The space Hσm is the completion of {ϕCc(Rd):divϕ=0} under the normϕHσm:=supxRdϕ(x)+|β|=1msupxRdDβϕ(x). The space C([0,T];Hσm) of continuous functions ψ is equipped withψC([0,T];Hσm):=sups[0,T]ψ(s)Hσm.

Section snippets

Forward-backward stochastic differential equations

Diffusion processes are linked to PDEs through the celebrated Feynman-Kac formula. The theory of Itô SDEs together Backward SDEs constitute systems of FBSDEs that are connected with deterministic nonlinear PDEs through a nonlinear Feynman-Kac formula, generalizing the classical Feynman-Kac formula (see e.g. [36], [37], [38]).

Let T>0, t[0,T) and xRd be fixed. Consider the FBSDEs on [t,T] of the form{Xs=x+tsb(r,Xr,Yr,Zr)dr+tsσ(r,Xr,Yr)dWr,Ys=g(XT)+sTh(r,Xr,Yr,Zr)drsTZrdWr, where b:[0,T]×Rd×

Numerical methodology

In this section, we specify some probabilistic algorithms for the numerical solution of FBSDEs. The stochastic representations for the solutions of the Burgers equation and the incompressible Navier-Stokes equations follow from systems (6) and (10), respectively. In the first case, we consider the approach by F. Delarue and S. Menozzi [15], [16], and then we incorporate quantization as a control variate variable to reduce the variance of the Monte Carlo estimation of the expected values. Then,

Numerical tests

Let u(t,x)=(u1(t,x),,ud(t,x)) be the exact velocity field that solves the incompressible Navier-Stokes equations (1). Since the equivalent backward form (8), by abuse of notation, we denote in the same way u¯N(t,x)=(u¯1N(t,x),,u¯dN(t,x)) the estimation of u(t,x) obtained from Algorithm 3.4 i.e. by means of Algorithm 3.3 with QN estimated as in (20). As in Algorithm 3.1, we only consider discretization-steps δ,h(0,1) such that δ<h. The pressure term is computed with step sizes τ(0,1) to

Conclusions

Forward-backward SDEs driven by Brownian motion are linked to nonlinear PDEs by means of the Feynman-Kac formula. The deterministic solution is interpreted as the conditional expectation of a diffusion process, and a stochastic algorithm to compute estimations is deduced from its probabilistic representation. A novel system of FBSDEs due to F. Delbaen, J. Qiu and S. Tang [14] introduces a probabilistic approach associated to the incompressible Navier-Stokes equations in dimensions d=2,3. The

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The authors thank editors and anonymous referees for their valuable comments and suggestions on the manuscript.

H. Mardones González gratefully acknowledges to the Institute Élie Cartan de Lorraine, France, for their hospitality during his research and the Departamento de Ingeniería Matemática at the Universidad de Concepción, Chile, for software facilities. The author thanks the funding of the Becas Chile, CONICYT grant 75140048, Chile, and the partial support by research team TOSCA, Inria

References (68)

  • G.G. Stokes

    On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids

    Trans. Camb. Phil. Soc.

    (1849)
  • L. Euler

    Principes généraux du mouvement des fluides

    Mém. Acad. Roy. Sci. Berlin

    (1757)
  • C.L. Fefferman

    Existence and smoothness of the Navier-Stokes equation

  • C. Foias et al.

    Navier-Stokes Equations and Turbulence

    (2001)
  • J.-Y. Chemin et al.

    Mathematical Geophysics - An Introduction to Rotating Fluids and the Navier-Stokes Equations

    (2006)
  • L. Tartar

    An Introduction to Navier-Stokes Equation and Oceanography

    (2006)
  • K. Itô

    Stochastic integral

    Proc. Imp. Acad. (Tokyo)

    (1944)
  • K. Itô

    On a stochastic integral equation

    Proc. Jpn. Acad.

    (1946)
  • R.P. Feynman

    Space-time approach to non-relativistic quantum mechanics

    Rev. Mod. Phys.

    (1948)
  • M. Kac

    On distributions of certain Wiener functionals

    Trans. Am. Math. Soc.

    (1949)
  • F. Delarue et al.

    A forward-backward stochastic algorithm for quasi-linear PDEs

    Ann. Appl. Probab.

    (2006)
  • F. Delarue et al.

    An interpolated stochastic algorithm for quasi-linear PDEs

    Math. Comput.

    (2008)
  • A.J. Chorin

    Numerical study of slightly viscous flow

    J. Fluid Mech.

    (1973)
  • A.J. Chorin

    Vorticity and Turbulence

    (1998)
  • S. Méléard

    Stochastic particle approximations for two-dimensional Navier-Stokes equations

  • Y. Le Jan et al.

    Stochastic cascades and 3-dimensional Navier-Stokes equations

    Probab. Theory Relat. Fields

    (1997)
  • S. Benachour et al.

    Branching process associated with 2d-Navier Stokes equation

    Rev. Mat. Iberoam.

    (2001)
  • R.N. Bhattacharya et al.

    Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations

    Trans. Am. Math. Soc.

    (2003)
  • M. Ossiander

    A probabilistic representation of solutions of the incompressible Navier-Stokes equations in R3

    Probab. Theory Relat. Fields

    (2005)
  • E.C. Waymire

    Probability & incompressible Navier-Stokes equations: an overview of some recent developments

    Probab. Surv.

    (2005)
  • P. Constantin et al.

    A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations

    Commun. Pure Appl. Math.

    (2008)
  • G. Iyer et al.

    A stochastic-Lagrangian particle system for the Navier-Stokes equations

    Nonlinearity

    (2008)
  • X. Zhang

    A stochastic representation for backward incompressible Navier-Stokes equations

    Probab. Theory Relat. Fields

    (2010)
  • D. Talay et al.

    A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations

    Ann. Appl. Probab.

    (2003)
  • View full text