A forward-backward probabilistic algorithm for the incompressible Navier-Stokes equations
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 (for space dimension ) where is a fixed time, 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 and pressure defined for each time and position . It involves a nonlinear convective term , a diffusion term and the incompressibility condition . 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 , 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 . With and , the process is solution to the FBSDEs where 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 . Under appropriate conditions, the pressure term p is interpreted by solving a Poisson problem 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 .
- •
For each , we set
- •
For each and , we denote by its transpose matrix while represents the identity matrix. The matrix is said to be elliptic if there exists such that The set of real valued symmetric non-negative matrices is
- •
The symbol “≈” means “is numerically approximated by”.
- •
The symbol means the order of estimation. For example, means that is depending on and less than , with constant K independent of δ. From now on stands for constants independent on the discretization parameters.
- •
The indicator function of a set A is .
- •
In general, we use the Einstein summation convention through this paper.
- •
The space , or simply as , of k-continuously differentiable functions with α Hölder continuity for each k-order partial derivatives equipped with the norm The multi-index notation is considered.
- •
For and we denote by the space of continuous functions ψ with norm Similarly we have the space of continuous functions with
- •
Let the space of predictable stochastic processes Ψ with values in such that
- •
Let be the set of all -progressively measurable stochastic processes Ψ with continuous trajectories taking values in such that
- •
For and we denote by and the usual -valued Lebesgue and Sobolev spaces on , respectively. When we just write and , or when .
- •
The space is the completion of under the norm The space of continuous functions ψ is equipped with
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 , and be fixed. Consider the FBSDEs on of the form where
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 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 the estimation of obtained from Algorithm 3.4 i.e. by means of Algorithm 3.3 with estimated as in (20). As in Algorithm 3.1, we only consider discretization-steps such that . The pressure term is computed with step sizes 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 . 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)
- et al.
Adapted solution of a backward stochastic differential equation
Syst. Control Lett.
(1990) A mathematical model illustrating the theory of turbulence
Adv. Appl. Mech.
(1948)- et al.
Forward-backward stochastic differential systems associated to Navier-Stokes equations in the whole space
Stoch. Process. Appl.
(2015) - et al.
Navier-Stokes equations and forward-backward SDEs on the group of diffeomorphisms of a torus
Stoch. Process. Appl.
(2009) On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case
Stoch. Process. Appl.
(2002)A numerical method for solving incompressible viscous flow problems
J. Comput. Phys.
(1967)Efficient Monte Carlo simulation for integral functionals of Brownian motion
J. Complex.
(2014)- et al.
A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes
J. Comput. Phys.
(2016) - et al.
Calculating effective diffusivities in the limit of vanishing molecular diffusion
J. Comput. Phys.
(2009) Mémoire sur les lois du mouvement des fluides
Mem. Acad. Sci. Inst. France
(1822)
On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids
Trans. Camb. Phil. Soc.
Principes généraux du mouvement des fluides
Mém. Acad. Roy. Sci. Berlin
Existence and smoothness of the Navier-Stokes equation
Navier-Stokes Equations and Turbulence
Mathematical Geophysics - An Introduction to Rotating Fluids and the Navier-Stokes Equations
An Introduction to Navier-Stokes Equation and Oceanography
Stochastic integral
Proc. Imp. Acad. (Tokyo)
On a stochastic integral equation
Proc. Jpn. Acad.
Space-time approach to non-relativistic quantum mechanics
Rev. Mod. Phys.
On distributions of certain Wiener functionals
Trans. Am. Math. Soc.
A forward-backward stochastic algorithm for quasi-linear PDEs
Ann. Appl. Probab.
An interpolated stochastic algorithm for quasi-linear PDEs
Math. Comput.
Numerical study of slightly viscous flow
J. Fluid Mech.
Vorticity and Turbulence
Stochastic particle approximations for two-dimensional Navier-Stokes equations
Stochastic cascades and 3-dimensional Navier-Stokes equations
Probab. Theory Relat. Fields
Branching process associated with 2d-Navier Stokes equation
Rev. Mat. Iberoam.
Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations
Trans. Am. Math. Soc.
A probabilistic representation of solutions of the incompressible Navier-Stokes equations in
Probab. Theory Relat. Fields
Probability & incompressible Navier-Stokes equations: an overview of some recent developments
Probab. Surv.
A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations
Commun. Pure Appl. Math.
A stochastic-Lagrangian particle system for the Navier-Stokes equations
Nonlinearity
A stochastic representation for backward incompressible Navier-Stokes equations
Probab. Theory Relat. Fields
A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations
Ann. Appl. Probab.
Cited by (4)
Numerical Solution of the Incompressible Navier-Stokes Equation by a Deep Branching Algorithm
2023, Communications in Computational PhysicsNew coal-charging method: analysis of force characteristics and velocity distribution of coal particles with dense phase continuous flow in cylinder
2022, Energy Sources, Part A: Recovery, Utilization and Environmental Effects