Continuum Mech. Thermodyn. 4 (1992) 247-267 C0ntinuum Mechanics T h and ] . erm0uynamlcs �9 Springer-Verlag 1992

A stochastic approach to computational fluid dynamics

H. P. Breuer and E Petruccione

In this paper a new approach to computational fluid dynamics is intro- duced in which the fluid is regarded as a stochastic dynamical system. The velocity of the fluid is related to a stochastic process governed by an ap- propriate master equation acting in a discrete phase space. The method is explained by means of (1 + 1)-dimensional flow phenomena. It is shown that the stochastic approach naturally leads to transparent numerical algorithms for stochastic simulations of fluid motion. By simulating plane Poiseuille flow it is demonstrated that the probabilistic approach yields a correct description of laminar fluid motion. Furthermore, soliton-like and shock wave solutions of Burgers' equation are generated by stochastic simulations of the underlying stochastic process.

1 Introduction

The basic equations describing the dynamics of fluid motion, i.e. the Navier- Stokes equation, have been known for a long time. Nevertheless, the integra- tion of these partial differential equations remains a challenging problem in spite of the increasing computational capabilities available today. For this reason, we suggest in this paper a new approach to fluid dynamics which treats the fluid as a stochastic dynamical system governed by a discrete master equa- tion. Thereby this approach avoids a mathematical formulation in terms of deterministic partial differential equations for macroscopic fields.

One of the central ideas of our approach is to relate the velocity field of the fluid to an appropriately chosen discrete stochastic process. Once a defini- tion of the phase space, i.e. the space of states of the fluid, has been given, a master equation governing the stochastic process can be formulated. On the other hand, the velocity which appears as a field variable in the Navier-Stokes equation is then interpreted as the expectation value of the stochastic process defined by the master equation. This transition from random variables to ex- pectation values provides the connection between the stochastic and the macroscopic description and applies also to variables other than the velocity.

248 ~" -

The great advantage of a r space of states is the fact tht efficient numerical implement : ,or the simulation of stochastic p _~,~Llons [1]. These methods are based on the num . . . . . . . o~ realizations of the stochastic process and the subsequent e,,~,uation of physical quantities as statistical averages. It is important to note that by stochastic simulation we do no t mean the solution of the master equation as an ordinary differential equation but rather the direct simulation of the underlying process. The stochastic simula- tion methods have already been successfully applied to the investigation of complex polymerisation reactions [2, 3].

In this paper we introduce this program of formulating the dynamics of fluids in terms of stochastic processes. We suggest to call this approach Hydrostochas- tics to stress the stochastic content of the theory. Here, in order to keep the presen- tation as simple as possible we only explain the (1+ 1)-dimensional version of our theory. The generalization to (2 +1) and (3 +l)-dimensional flow phenomena is straightforward and will be given elsewhere. In this work we consider typical flow situations which can be reduced to ( l+l)-dimensions by symmetry arguments. Basically, these flow phenomena are characterized by the presence of a viscosity term and of a constant pressure gradient. Furthermore, in order to show that also nonlinear inertial terms can be handled within our theory we study the (1+1)- dimensional Burgers equation. This equation may be regarded as the simplest ver- sion of the Navier-Stokes equation of a compressible fluid in one space dimen- sion; however, it exhibits many characteristic features of real fluids, since it con- tains a nonlinear inertial term as well as a viscous diffusion term. A further reason for choosing Burgers' equation is the fact that the initial and boundary value problem can be solved analytically in closed form. Thus, this equation may serve as a nonlinear test equation for numerical algorithms.

Our paper is organized as follows. In Sec. 2 we give a heuristic construction of the master equation of our approach. Furthermore, we show how to impose arbitrary initial and boundary conditions in the context of Hydrostochastics. Finally, in this section we sketch how stochastic simulations are performed. Sec. 3 is devoted to the stochastic simulation of processes governed by master equations. As an example of fluid motion in ( l+l)-dimensions we simulate laminar plane Poiseuille flow. Furthermore, we show that a straightforward simulation of the master equation replacing Burgers' equation naturally generates, e.g., nonlinear soliton-like and shock wave solutions. Finally, in Sec. 4 we draw our conclusions.

2 Hydrostochastics

In this section we first define the space of states of the fluid. Once the accessi- ble states of the fluid are known it is possible to construct a master equation governing the probabilistic time evolution of the system. Furthermore, we demonstrate how the deterministic equations o f motion emerge from the master equation as the equations governing the time evolution of the expecta- tion value of the random velocity.

A stochastic approach to computational fluid dynamics 249

2.1 The definition of phase space

The first step of the construction of an appropriate phase space consists in the division of physical space into a number of cells centered at the points x;~ which we label by an integer-valued index 2 fi Z. For the purpose of general (three dimensional) considerations it suffices to use cubic cells of size 6l 3; in practice, shape and size of the cells can be adapted to the confining geometry of the flow. Note, that in this paper the cells are just one-dimen- sional intervals of length Ol since we are considering one space dimensional examples only. Depending on the physical situation under study the number of cells may be finite or infinite. In the stochastic simulation this number is, of course, always finite and fluids of practically infinite size have to be modell- ed by imposing suitable boundary conditions (see below).

According to our general philosophy outlined in Sec. I we now interpret the velocity field u (x, t) appearing in the Navier-Stokes equation as an expectation value of a discrete stochastic process N~(t) ~ Z, i.e. we define

u(x z, t) = ~u(N~(t)). (1)

The above equation provides the connection between the macroscopic and the stochastic description of fluid motion. Within the stochastic description the velocity is a time-dependent random variable N~, i.e. a stochastic process governed by a master equation to be defined below. On the other hand, the velocity on a macroscopic scale, that is on the scale accessible to standard ex- perimental observation, is given by an expectation value, and therefore obeys the deterministic Navier-Stokes equation. The velocity unit Ou has been in- troduced in order to obtain a discrete stochastic process N~. Thus it represents the size of the smallest possible changes of the state of the fluid in the discretized phase space. This means that within our description a positive value Ou. N~ of the random velocity in the cell 2 can be interpreted as the presence of N~ velocity particles each carrying the velocity Ou. Cor- respondingly, a negative value Ou. N~ is to be interpreted as the presence of INk] antiparticles of velocity. Defining the positive and negative part of N~ as

[Nu ~, N ~ > 0 )~ f 0 , N ~ > 0 N 2 u + : : = , Nu-:= = ,

O, N ~ < O ~-N~, N ~ < O (2)

we write:

N~ = N~+ + N~_. (3)

Thus we see that within our description the mesoscopic state of the fluid may be viewed as a many velocity particle state and is completely fixed by specify- ing the number N~ of velocity particles in each cell 2. Formally, the resulting phase space may be written as

N ~ r = {( ~)~z JN~ ~ z}. (4)

250 H.P. Breuer, E Petruccione

2.2 The construction of the master equation

In order to obtain the master equation of Hydrostochastics we proceed as follows: We show explicitly how stochastic processes can be constructed whose expectation values obey the corresponding terms in the Navier-Stokes equation. These stochastic processes are defined by a master equation which is a time evolution equation for the joint probability distribution P = P(NX.; t) in the phase space F. From this probability distribution the expectation value for an arbitrary observable ~Y may be evaluated according to

(~) -'- E ~P([N~u}; t). (5)

2.2.1 Deterministic terms as Poisson processes

To begin with let us consider the most simple situation, i.e. the Navier-Stokes equation in one dimension for an ideal, incompressible fluid

Ou Op - . ( 6 )

Ot Ox

Since the continuity equation in one dimension reads

Ou - - = O , ( 7 ) Ox

neither u nor Op/Ox can depend on the spatial coordinate. Therefore, assuming that Op/Ox does not depend on time equation (6) describes a uniform linear growth of the velocity u with time.

In order to model this deterministic behaviour within our mesoscopic pic- ture we regard u( t ) as the expectation value of an integer valued Poisson ran- dom process [4] Nu(t) , i.e. we define

u( t) = fiu(Nu( t) ). (8)

The Poisson process underlying the partial differential equation (6) is defined by the following master equation governing the time evolution of the probabili- ty distribution P ( N u, t)

OP - - = k ( P ( N u - 1, t) - P ( N u, t ) ) . (9) Ot

For the corresponding expectation value one easily obtains

u( t ) = 6u(Nu( t ) ) = ~u kt. (10)

The transition rate k has to be chosen such that this expectation value shows the correct time dependence, i.e.

a O p k = 01)

6u Ox"

A stochastic approach to computational fluid dynamics 251

The important fact to be noticed here is that the Poisson process has, in con- tradistinction to other stochastic processes, a deterministic limit. In fact, in the limit

fu --+ 0 and fuN, = const. , (12)

we obtain

P ( N u ' t ) ~ f ( N u f u - k f u t ) = f ( Nufu+ Op " (13)

Obviously, the above reasoning can be generalized to the case of a space dependent pressure gradient. In such a situation, having divided space into cells one has to define in each cell an independent Poisson process N~. Hence, the generalization of the master equation (9) for the joint probability distribution

P = P(N~; t)

reads

(14)

OP - E kZ(E~-I - 1 )P , (15)

Ot 2

where, again, k ~ denotes the space dependent negative pressure gradient di- vided by flu. In the above equation we introduced the shift operators Ea defined by

E~F( . . . . N~ . . . . ) = F( . . . . N~ + 1 . . . . )

E ; 1 F ( . . . . N~ . . . . ) = F ( . . . . N~ - 1 , . . . ) , (16)

where F is an arbitrary function of the random variables. In the simplest cases we are going to consider in this paper, the transition rates k ~ are independent of 2.

Within our many-particle interpretation the stochastic process defined by Eq. (15) may be understood as a creation of velocity particles independently in each cell. Therefore, k ~ is the creation rate in the cell 2. Of course, we im= plicitly assumed in Eq. (15) that the negative pressure gradient k~ is positive. It should be clear how to deal with negative k~: the latter are to be inter- preted as creation rates of antiparticles of velocity. Hence, the complete master equation reads

OP at - E k~ (Ex--1 - 1 ) P + ~ - k X ( E a - 1 ) P =: dpP, (17)

kx>0 kx<0

where we defined the "pressure operator" d p acting on the probability distribution.

252 H. E Breuer, F. Petruccione

2.2.2 Viscosity as a many-particle diffusion process

In this section we are going to describe viscous fluids within our stochastic approach. In the simplest cases, the Navier-Stokes equation containing only viscous forces reads

Ou 1 02u

Ot - R Ox 2" (18)

As in the preceeding section we want to define a stochastic process NZ,(t) in such a way that its expectation value

u(x z, t) = ~u(N~u(t)) (19)

obeys the diffusion-like equation (18). It is well-known that the probability distribution 4~z(t), )~ ~ Z, of a one

particle random walk fulfills in the continuous limit a diffusion equation. More precisely, let us consider a continuous time random walk in one dimen- sion. The probability distribution ~b~(t) obeys the following master equation

- d. (~bx+l - 2~b~ + ~bx_l), d = const. . (20) Ot

It is easy to see from the above equation that in the limit of infinitesimal small random steps,

~bz(t) -~ 6(x = Ol. 2, t) , (21)

the probability distribution q~(x, t) is a solution of the diffusion equation

0th _ D - - 02q5 D : = lira (t~l 2 d) . (22) Ot Ox 2 ' ~l~o

The formal analogy of the diffusion equation (22) with the one dimensional Navier-Stokes equation (18) is obvious. However, there is a fundamental dif- ference between these two equations. The diffusion equation (22) describes the time evolution of a probability distribution which is normalized and positive definite. In contrast, the one dimensional Navier-Stokes equation (18) is an equation for an expectation value which might be negative and not nor- malizable. Thus, the stochastic process underlying (18) can not be described by a one particle random walk.

Therefore, one has to leave the one particle picture. To this end, we con- sider a collection of independent velocity particles each of which is governed by Eq. (20). The state of the resulting collective system [4] is characterized by the set of numbers N~ of velocity particles in all cells ft. Thus, in this many- particle picture one disregards the identity of the individual particles and is merely interested in the occupation numbers N~ of particles in each cell. (Note, that in the one-particle picture the state of the system is completely specified by giving the location )~ of the particle.)

In the framework of the many particle picture the one-particle density 4z(t) is replaced by the many particle probability distribution P(N~; t). The

A stochastic approach to computational fluid dynamics 253

master equation defining the time evolution of P(N~.; t) can be obtained from the one-particle master equation by multiplying the one-particle transi- tion rates with the occupation number N~ of cell 2; hence we obtain:

OP 1 Ot - ROl 2 ~ [(E;11 E~ - 1) N~ + (E;+11 E2 - 1) Null P. (23)

2

where fil denotes the cell size. Since the expectation value of N~ is a statistical guess of the one-particle density at the point 2 the velocity field (19) obeys the diffusion equation (18) as required.

Until now we implicitly restricted our attention to the collective random walk of positive velocity particles. Obviously, to be consistent with our general picture of particles and antiparticles of velocity we have to generalize the master equation describing viscous fluids in order to allow for the correct treatment of antiparticles of velocity. Since a one-particle jump to the right is equivalent to a one-antiparticle jump to the left and vice versa, the master equation (23) has to be modified in the following way: In order to describe the diffusion of antiparticles the numbers N~ = N~_ are replaced by their ab- solute values tN~[ = -N~_ and the shift operators EZ.11Ea by E2• 1EF 1. Con- sequently, if both particles and antiparticles of velocity are present the master equation can be written:

OP

Ot

1 - - (EJ_IE 2 - 1) Nu~+ + (EZ+11E~ - 1) N~u+]P RSI 2 ~ [ -1

1 Rill 2 E [(E'~-IE~-I - 1) Nu ~_ + (Ez+ 1E~ -1 -

Z

= : ddP,

where we introduced the "diffusion operator" S d.

1) N~u_] P

(24)

2.2.3 The nonlinear convection term

Up to now, we have shown how Hydrostochastics deals with a pressure gra- dient and viscous diffusion. However, it is of great importance to include within the stochastic theory nonlinear interactions which, for example, enter the Navier-Stokes equation through the nonlinear inertial term (ft. ~) ft. It is a remarkable fact that interpreting the inertial term as a nonlinear convection term a stochastic interpretation in terms of one-particle jumps can be found. As will be shown here, again, the many-particle picture is absolutely necessary.

It is a well known fact that the most simple one-dimensional equation describing both nonlinear propagation and viscous diffusion is Burgers' equa- tion [5], Ou Ou 1 02u - - + u - ( 2 5 ) at Ox R Ox z"

Burgers' equation has been used as a simple model of turbulent motion. Fur- thermore, it admits solutions representing nonlinear wave motion [6]. Note that

254 H.P. Breuer, F. Petruccione

the term uOu/Ox can be considered to be the one-dimensional version of the inertial term ( i f - ~ ) a in the Navier-Stokes equation.

As before we are looking for the stochastic process Nau(t) the expectation value of which obeys Burgers' equation. Obviously, the stochastic process underlying the nonlinear convection term is fundamentally different from the previously introduced stochastic processes: It is neither a random walk of a collection of independent particles nor a Poisson creation process. The con- struction of the Hydrostochastic form of the convection term will now be given heuristically within the many particle picture.

Let us consider what happens in a specific cell 2 occupied by N~ __> 0 velocity particles. Within the stochastic approach convection may be modelled as a jump process from cell 2 to cell (2 + 1). Thus, the stochastic process is completely specified by giving the transition rates for these elementary jumps. The probability for the jump of a specific particle situated in cell 2 is proportional to the velocity at 2 and, hence, proportional to the number of velocity particles in cell 2. Consequently, the total transition rate is propor- tional to the number of pairs of velocity particles in the cell 2, i.e. to Nu z. (N~- 1)/2. On dimensional grounds the proportionality factor is found to be J u / J l (note that transition rates have the dimension of an inverse time) and we obtain the following master equation for the convection of velocity particles:

OP Ju 1 ~ Ot - Jl E (E~-+11Ez - l) 2 Nu(Nu - 1)P. (26)

In Sec. II.C we demonstrate how Burgers' equation emerges from the master equation formulation as an equation for the expectation values. It can be shown that the above form of the master equation for the convection term leads to a discretization of the differential operator uO/Ox which is of order Jl. In order to obtain a discretization of this differential operator which is of order Jl 2 the following symmetrized form of the master equation will be us-

_ 1 Ju (E~-+11E~ - 1) 2 N~u(N~ - 1) P Ot 2 J l

1 ~ ~ p~ q- E (E~-I Ez-1 -- 1) ~- N u ( N u - 1)

) ). = : ~r (27)

where we abbreviated the effect of the right hand side by defining the "convec- tion operator" 5r c.

Constructing Eq. (27) we assumed that N~u _>_ 0. The general master equa- tion describing both the presence of velocity particles and antiparticles can now easily be derived. Let us assume, that cell 2 contains ]NuZl =-NZu_ anti- particles of velocity. The elementary jumps of the antiparticles can be obtained from those of the particles of velocity by simply reversing the directions of the jumps, i.e. all antiparticles jump to the left. However, since the process

ed:

OP

A stochastic approach to computational fluid dynamics 255

of an antiparticle jumping to the left is identical to the process of a particle jumping to the right, the above master equation describes the combined con- vection of particles and antiparticles of velocity if N, ~ is replaced by [N~ [.

2.3 The master equation o f Hydrostochastics

Having discussed how to interpret the various terms occuring in the equations of fluid dynamics as stochastic processes let us now combine these processes to obtain the unified master equation formulation of Hydrostochastics. Ob- viously, the combined effect of different terms can be obtained by just adding the corresponding operators. For example, the master equation describing a pressure driven viscous fluid in one space dimension reads

OP ot = I @ + e , (28)

where the pressure and diffusion operators @ and J d have been defined in Eqs. (15) and (24). As another example let us construct the master equation corresponding to Burgers' equation (25); we obtain from Eqs. (24) and (27):

OP

Ot

1 -- Rt~l 2 E [ (E~_1-1 Ex - 1) Nu+X + (E~-+I1 E;~ - 1) N,z+] P

2

1 - - N u - ] P R g l 2 E [(Ej'-I E;1 1) Nu ~_ + (Ej~+I Ex --1 1) ;~

1 (E;11E - 1) 2 IN I(IN I- 1)P +2 1 PI + E (E;1E;~-I - 1) 2 [N~I (IN.~I - 1) . (29)

),

Although the above equation may look rather complicated it allows for a transparent physical interpretation within the many-particle picture. Recall, that the system is characterized by giving the numbers N~ of velocity particles in each cell of the discretized space. The above master equation defines the possible transitions between the states in the phase space F and the correspond- ing transition rates. These transitions solely consist of one particle jumps to neighbouring cells each particle carrying the amount gu of velocity. The latter fact turns out to be of fundamental importance as it makes possible the con- struction of efficient numerical schemes which simulate the stochastic process underlying the master equation (see Sec. 2.5). Furthermore, the nonlinear con- vection term in the master equation is responsible for the fact that in the many-velocity particle system a nonlinear self-interaction is present,

256 H.P. Breuer, E Petruccione

Let us now demonstrate in some detail how the macroscopic equations may be derived from the corresponding master equations. As an example we con- sider the master equation (29) corresponding to Burgers' equation (25). Generally, the time derivative of the expectation value ua = (OuN~) may be written as the expectation value of a commutator

Ot (30)

From the properties of the shift operators Ez it follows that (recall that Nu =Nu+ +N~_)

t~bt ,,h12+ 1 __2Nu~+ +Nu~+l 4_ jVz+ 1 -2N~_ +N~_ ) ([6uN~u, del ) = R 6l 2 v, u + ---. u-

t~U ,~N2 + 1 2 2 -1 - R M 2 ,~., - 2 N u + N u ). (31)

Evaluating the commutator of the operators N~ and ~ we obtain neglecting terms of order 1/N~

~u 2 ([6uNZu, d c l ) - ((N.u2+l) 2 - (Nu~-l)2). (32)

4dl

Since for sufficiently small 0u the number of velocity particles N~ becomes large one expects that fluctuations are small and, therefore, that the approx- imation

((N~) 2) = (N~u)2 (33)

holds to a sufficient degree of accuracy. Eq. (30) may then be written as an equation for u~,

0U 2 1 U2+l - - U]_ 1 1 U,~+I -- 2U,~ + R;t_ 1 - - + - ( 3 4 ) Ot 2 2(5l R 6l 2

Obviously, this equation is nothing but the discretized version of the Burgers equation (25) which emerges in the continuum limit ~l-~ 0.

As has been mentioned in Sec. 2.2.3 the form (27) of the convection term leads - in the equation for the expectation value (Su(N~) - to an approx- imation of the differential operator uO/Ox which is of order (~/)2.

In deriving the above macroscopic equation for the expectation value ux we neglected, of course, all higher moments of the stochastic process N~. As is well-known the above master equation, defining a nonlinear stochastic pro- cess, leads to an infinite dimensional system of coupled differential equations for the moments. Thus, in order to derive more rigorously from our master equation the macroscopic equation and to investigate the dynamics and in- fluence of fluctuations one has to employ a more systematic method. Such a method is provided by the well-known O-expansion [4]. Applying this expan- sion to the master equations of Hydrostochastics reveals that, in fact, the

A stochastic approach to computational fluid dynamics 257

macroscopic equations are equivalent to the equations of fluid dynamics. Fur- thermore, it can be shown that within the linear noise approximation the fluc- tuations superimposed on the macroscopic dynamics can be identified with those fluctuations derived from the theory of fluctuating hydrodynamics [7]. Thus, the stochastic dynamics implied by our master equation formulation can, indeed, be given a clear physical interpretation. However, these consider- ations are beyond the scope of this paper and will be presented elsewhere [8].

In this work, we confine ourselves to the treatment of (1 + 1)-dimensional fluid dynamical problems. The generalization of the master equation to two and three space dimensions can be performed within the same framework of the theory presented above.

2.4 Boundary and initial conditions

In the preceeding sections we have derived the fundamental master equations of our theory. As one might expect the correct implementation of boundary conditions is decisive for the description of fluid motion within the context of Hydrostochastics as it is within the formulation of fluid dynamics with the help of partial differential equations. In Hydrostochastics the correct boundary conditions have to be imposed on the master equation, e.g. eq. (29), describing fluid motion.

In this section we want to describe how to take care of a boundary separating a viscous fluid and a solid surface. Typically, there exist two types of boundary conditions. Firstly, at fixed solid surfaces the no-slip boundary conditions require that the velocity vanishes. Secondly, for moving boundaries the no-slip condition requires the velocity of the fluid at the solid surface to be equal to that of the boundary.

The boundary condition at fixed solid surfaces

/~[ Boundary = 0 (35)

results from the fact that adhesion forces create an infinitesimal fluid layer of zero velocity along the solid surface. As we already described in detail one of the fundamental steps in the formulation of Hydrostochastics is the parametri- zation of the fluid bulk in cells 2. The infinitesimal zero velocity fluid layer can thus be modelled by introducing a corresponding layer of cells. The cells within this layer act as sinks of diffusing velocity particles. In practice, this means that velocity particles jumping into the layer cells disappear. On the other hand, this implies that no velocity particle can leave the layer cells since the velocity of the boundary layer is zero.

In the case moving solid surfaces the boundary condition reads

b~j Boundary = /Q, (36)

where 0 is the velocity of the moving solid surface. As in the case of the solid surface at rest we again introduce a layer of boundary cells. Now, the layer cells are supposed to move with the constant velocity U.. In other words, this means that in each layer cell the number of velocity particles is kept constant. If for example the solid surface moves with constant velocity U in the x direc-

258 H.P. Breuer, E Petruccione

tion, the number of velocity particles Nu in each layer cells has to be chosen such that U = 6u(Nu)= 0uN,. There is thus a fundamental difference to the case of a fixed boundary: Now velocity particles may leave the layer cells. However, in the latter case the velocity of the layer ceils or equivalently the number of velocity particles in these cells is kept constant. As for the fixed boundary, particles diffusing from the neighboring cells into the layer ceils disappear.

For the investigation of the properties of the fluid bulk it might be necessary to use periodic boundary conditions. Also the implementation of these boundary conditions does not cause any problem within our approach, since all transitions have been reduced to one-step processes (see Sec. 3).

As far as initial conditions are concerned, let us assume that the initial velocity in cell X is u0. This initial condition can be realized by fixing in an appropriate way the number of velocity particles in the cell under considera- tion. In our example it is sufficient to put at the beginning of the simulation N O = Uo/OU velocity particles in cell )~.

2.5 Stochastic Simulation

That the master equation formulation within a discretized phase space natural- ly leads to efficient numerical algorithms is one of the major advantages of our approach. In this section we present a brief description of the method of simulation of stochastic processes governed by master equations [1]. The method of stochastic simulation is particularly well suited for the numerical investigation of high dimensional systems [2, 3]. This method is applied in the next section to simulate the stochastic processes of our theory.

Basically, the method of stochastic simulation produces realizations NZu(t) of the underlying stochastic process. From a sufficiently large number of these realizations one then can obtain physical quantities as statistical averages.

The generation of a realization proceeds along the following lines: Assum- ing that the system is in a specific state N~u(to) at time to one can infer from the master equation the rate for the next transition to occur. This total transi- tion rate W(NZ~(to)) is, of course, the sum of the rates of all possible jumps of velocity particles and antiparticles. From the total transition rate W one can evaluate the waiting time distribution, i.e. the probability distribution of the time interval dt the system remains in the state N~u(to). According to this distribution the random number dt can be obtained from a uniformly dis- tributed random number t/E [0, 1] by the following relation

1 d t - W(N~(to) ) lnr/. (37)

Thus, having determined the transition time to +dt one chooses a specific transition with a relative probability which is given by the ratio of the cor- responding one-particle transition rate to the total rate W(N~(to)). In this way a new state N~(to +dt) is found. Repeating this procedure yields a tra- jectory N~u(t), i.e. a realization of the stochastic process.

A stochastic approach to computational fluid dynamics 259

Having thus determined a sufficiently large number M of trajectories (N~(t))j, j = 1 , . . . , M, interesting observables can be evaluated as statistical averages; for example the velocity is computed as

0u M u(x~, t) = Ou(N~u(t))= ~ E (N~(t))j. (38)

j = l

3 E x a m p l e s in t w o s p a c e - t i m e d i m e n s i o n s

In this section we are going to apply Hydrostochastics to some typical ( l + l ) - dimensional flow situations. We compare the results of the simulations with well-known analytical results of Hydrodynamics.

3.1 Plane Poiseuille Flow

In this subsection we consider the laminar flow between two parallel plates subjected to a constant pressure gradient. We define the Reynolds number R with respect to the parameters

L = distance between the plates

U = maximal velocity of the stationary flow field

v = viscosity of the fluid, (39)

i.e.,

U.L R - (40)

V

We choose coordinates in such a way that the plates are located at x = 0 and x = 1. The flow is directed along the positive y-axis and we denote the y-com- ponent of the velocity by u = u(x, t).

The hydrodynamic description of this flow is given by the following form of the Navier-Stokes equation in two space-time dimensions:

Ou dp 1 02u - + - (41)

3t dy R OX 2"

In the case under study we assume the fluid to be initially at rest

u(x, 0) = 0, (42)

and according to the coordinates chosen we have the boundary conditions

u(0, t) = u(1, t) = 0. (43)

This initial and boundary value problem can be solved by standard means and yields for the time-dependent velocity field:

( - l r 2 n 2 t ) R 32 sin(~znx) exp . (44) u(x, t) = 4x(I - x) - Z 793n ~ n = 1,3,5 . . . .

260 H. R Breuer, E Petruccione

Let us now demonstrate how Hydrostochastics treats the above flow phenomenon. To this end we first have to implement the initial and the boun- dary conditions in our simulation. The initial condition (42) means that we have to start our simulation with the configuration

N~(0) --- 0. (45)

The boundary condition of fixed solid planes is implemented in the following way. The planes are modeled by two additional cells. In these cells one keeps the number of velocity particles equal to zero. This means that no velocity particles can leave the boundary cells and enter the fluid bulk. On the other hand, if a bulk velocity particle diffuses into a boundary cell it disappears. Having imposed these boundary and initial conditions the stochastic simula- tion proceeds along the lines sketched in Sec. 2.5.

The results of such a simulation are presented in the next figures. Figure 1 displays the result of a stochastic simulation of plane Poiseuille flow with the initial value (42) and compares it with the analytical result (44). We plot the scaled velocity field u ( x , t) for three different values of the time (t = 0.2 tr, t = 1.0 tr, and t = 5.0 tr) measured in units of the relaxation time t r = R / z r 2. Fig. i illustrates the way the velocity profile changes its shape and, finally, reaches the stationary solution.

1,25

1.00

0.75-

0,50 -

0.25 -

0 0 0.1 0.2 0.3 0.4- 0.5 0.6 0,7 0.8 0.9 1.0

X .------',--

F i g . 1. The result of a stochastic simulation of plane Poiseuille flow corresponding to the initial and boundary conditions Eqs. (42) and (43). We display the velocity as a function of x for the three different times t 1 = 0 . 2 , t 2 = 1 . 0 , and t 3 = 5.0 measured in units of the relaxation time. Smooth curves: The analytical result according to Eq. (44). Symbols: The stochastic simulation employing 100 realizations and 20 cells. 6u was chosen to be 0.001

A stochastic approach to computational fluid dynamics 261

This example demonstrates that it is indeed possible to incorporate the ef- fects of the deterministic pressure gradient as well as that of the viscosity term in a unified stochastic interpretation of the Navier-Stokes equation. Further- more, we see that the zero boundary conditions can be implemented quite simply by regarding the solid walls at x = 0 and x = 1 as sinks for the diffus- ing velocity particles. Finally, the agreement between the simulation and the analytical curve is remarkably good during the whole time evolution. This reflects the fact that we are simulating a stochastic process and not merely a stationary or equilibrium state.

3.2 Nonlinear waves

Originally Burgers proposed the Eq. (25) as a simple one-dimensional model of homogeneous turbulence. The main features of the Navier-Stokes equations are retained in the above equation. The nonlinearity has the same structure as in the Navier-Stokes equation, and the dissipative term is also of the same type. Only the pressure term is missing, so that one has to expect a relaxation of turbulence with time. The model also lacks an equation of continuity, so that it describes in practice a one-dimensional compressible flow. The Burgers equation is particularly appealing because the analytical solutions of the initial value problem is known [6]. These solutions represent, for example, nonlinear wave solutions like shock waves and "humps". Thus, Burgers' equation is in- teresting as it makes possible the study of the interplay of nonlinear propaga- tion and viscous diffusion.

3.2.1 Shock waves

The first example we are going to treat is the shock wave solution of Burgers' equation. The shock wave solution is obtained for the following initial condi- tion

I1 , x < 0 (46) u ( x , O ) = u o ( x ) = O, x > 0 "

The diffusion and the convection of this initial step are described by the following time-dependent solution

1 u(x, t) = , (47)

1 + h(x, t) exp [R(x - t /2) /21

where the function h(x, t) is defined by

erfc ( - x / ~ x ) h(x, t) = (48)

erfc ( x/-R/ 4t (x - t ) )

The function erfc(x) is known as the conjugated error function, and is given by

2 erfc(x) - ] dy exp( -y2) . (49)

262

1.25 -

1.00

0.75 -

0.50 -

0.25 -

0 - 1 . 0 0 - 01.75 -01.50 -01,25 i

X :

H. P. Breuer, F. Petruccione

c n ~

0.25

\ o:5o 0'.75 1:oo

Fig. 2. T h e s tochas t i c s imu la t ion o f the shock wave so lu t ion o f Burgers ' e q u a t i o n cor- r e s p o n d i n g to the initial condition Eq. (46) for the three different times tl = 0.1, t2 = 0.3, and % = 0.9. Reynolds number: R = 100 Smooth curves: The analytical result according to Eq. (47). Symbols: The stochastic simulation using 10 realizations and 100 cells. 6u was chosen to be 0.001

To generate shock waves within the master equation approach we have to implement the initial conditions properly. In this case it is sufficient to choose the number of velocity particles in the cells at the beginning of the simulation in the following way. No velocity particles have to be in the cells with x > 0. In the cells with x < 0 one has to put N, = 1/6u velocity particles. The boundary conditions are realized by keeping fixed the number N, of particles in one boundary cell and by keeping the number of particles in the other boundary cell equal to zero.

In Figure 2 we show the time development of the shock wave as it is otain- ed from the simulation at three different times ta = 0.1, t 2 = 0 .3 , and % = 0.9, and compare it with the analytical solution (47) (solid line). The Reynolds number was chosen to be 100. As one can see in this figure the agreement is overwhelming. It might be important to notice that the simulation results are particularly good on the front of the shock wave.

3.2.2 Single hump

Another interesting solution of Burgers' equation is the single hump solu- tion. This solution may be obtained by considering the following initial condition

u(x, O) = Uo(X) = A~(x ) . (50)

A stochastic approach to computational fluid dynamics 263

The propagation of the single hump is described by the following exact solu- tion

[exp (RA/2) _ 1] exp ( _ R 2)

u(x, t ) = (51)

1 [ e x p ( R A / 2 ' - 1 ] e r f c ( ~ t x )

In the simulation the initial condition was realized by setting initially N~ = 0 for all but one cell ,1. = 0 for which N o was chosen such that A = 2, i.e. we have at t = 0 :

0(fi~u2~/) for / l . 0, = (52)

N"x(0) int , for /l = 0,

where int (y) denotes the integer part of y. In Figure 3 we show the result of the stochastic simulation together with the analytical result (51) (solid line) for three different times t = 0.01, t = 0.05, and t = 0.10. The Reynolds number was chosen to be R = 10. Keeping in mind that a singular initial condition is studied here, the agreement between the stochastic simulation and the analytical result is striking although only 201 cells have been used. Furthermore, it should be emphasized that we did not employ any special technique to obtain Fig. 3, nor did we perform any analytical pre-integration in order to smooth the singular initial condition, nor did we use any analytical a priori guess of the solution.

17.5 -

15.0 -

12.5 -

I 10.0-

7.5-

5.0-

2.5-

0 -1.00

O

'0'.75 -0'.50 0.25 0.50 075 1.~0 -o'.25 ; X

Fig. 3. The stochastic simulation of the single hump solution of Burgers' equation cor- responding to the initial condition Eq. (50). The velocity is plotted for the three different times t 1 = 0.01, t 2 = 0.05, and t 3 = 0.1. Reynolds number: R = 10. Smooth curves: The analytical result according to Eq. (51). Symbols: The stochastic simulation using 10 realizations and 201 cells. 5u was chosen to be 0.01

264 H.P. Breuer, F. Petruccione

3.2.3 Periodic Waves

Finally, we discuss another important class of solutions of Burgers' equation: Imposing periodic boundary conditions for the velocity field,

u ( x + 2, t) = u (x , t ) , (53)

the following solution of Burgers' equation is obtained [6]

1 E n n e x p [ - 4 ( t + 1) ( x - c t - n ) 2]

u ( x , t) = - - - ct . . . . . . . . . + c , (54) t + l [ R 2]

E nexp 4 ( t + l ) ( x - c t - n )

where the sums are extended over the even integers n = 0, • 2, • 4 . . . . . This solution represents a periodic array of shocks separated by smooth ramps mov- ing to the right with mean velocity

2

1 ~ dr u ( x , t) = c = const. . (55) 2 0

From the examples shown above it shuld be clear how to realize the initial conditions corresponding to the exact solution (54) at t = 0 in a stochastic simulation. We therefore only explain how to impose the periodic boundary

1 . 5 -

1 . 0 -

0 . 5 "~

T 0 -

- 0 . 5 -

- 1 . 0 - o'.2s o'.so o% .'oo 1 25 .'so 1.'75 2%

X " i.

Fig. 4. The stochastic simulation of a periodic wave solution of Burgers' equation with boundary condition (53), mean velocity c = 0.5, and Reynolds number R = 100. The velocity is plotted for the three different times t 1 = 0.0 (initial condition), t2 = 0.5, and t 3 = 1.0. Smooth curves: The analytical result according to Eq. (54). Symbols: The stochastic simulation employing 10 realizations and 100 cells; 5u = 12001

A stochastic approach to computational fluid dynamics 265

conditions. Recall that having divided the period ranging from x = 0 to x = 2 into a number of cells the whole time evolution according to the master equa- tion has been reduced to a succession of one particle jumps to neighbouring cells. Hence, periodic boundary conditions are nothing but the following rule: A particle in the first cell (at x = 0) which jumps to the left ends up in the last cell (at x = 2), likewise, a particle in the last cell which jumps to the right ends up in the first cell.

In Figure 4 we display the results of our stochastic simulation. We show the velocity for the three different times t~ = 0, t2 = 0.5, and t~ -- 1.0, where the first time tl corresponds to the initial condition. The Reynolds number was chosen to be R - - 100 and the mean velocity c = 0.5. The lesson to be learned from the excellent agreement between the simulation and the analytical result (note, that only 100 cells have been used!) is twofold: Firstly, the im- plementation of the periodic boundary conditions is obviously correct. Second- ly, note that the parameters have been fixed in such a way that the initial state contains particles as well as antiparticles of velocity. Thus, the figure proves our stochastic interpretation of positive and negative velocities to be correct.

4 Conclusions

In the present work we presented a new approach to fluid dynamical computa- tions. The basic idea behind our approach is that we are not concerned with deterministic time evolution equations, i.e. partial differential equations, but rather with stochastic processes governed by master equations. The interesting physical quantities are then evaluated by statistical means from a stochastic simulation.

Of course, statistical methods have previously been applied to the solution of boundary value problems of partial differential or integral equations. However, these Monte Carlo methods [9] are concerned with stationary solu- tions, whereas in our approach we describe and simulate stochastic processes, i.e. time-dependent probabilistic evolution. Other stochastic approaches to the stationary solution of nonlinear problems, e.g. to Burgers' equation [10], are - contrary to our approach - not working in a many-particle picture and therefore have to rely on a priori guesses of the solution.

The starting point of our reasoning was the interpretation of the Navier- Stokes equation as an equation for the mean value of a stochastic process. This stochastic process is defined by fixing an appropriate phase space F which contains all possible states of the fluid and by specifying the rates of the tran- sitions between the states. In our investigation it turns out that the stochastic state of the fluid is defined by giving the set of numbers (N~)z of velocity particles in each cell 2 of discretized space.

In order to find the stochastic process N~ we combine different stochastic processes the expectation values i~f which represent the different terms of the Navier-Stokes equation. It turns out, that this can be done in a consistent way on- ly if the state of the system is regarded as a collective system, i.e. as a many-veloci- ty particle state. In particular, in order to find the stochastic analogon of the nonlinear convection term this many-particle interpretation is indespensable.

266 H.P. Breuer, F. Petruccione

The practicability and efficiency of our approach has been demonstrated with the help of some typical two space-time dimensional flow situations. We demonstrated that our probabilistic theory indeed yields a correct description of laminar fluid motion paradigmatic for deterministic time evolution. The stochastic simulation of nonlinear convection was illustrated by means of the one-dimensional Burgers equation. As we have seen it is quite easy to obtain localized soliton like and shock wave solutions of this equation from a stochastic simulation. Thus, Hydrostochastics leads to a stochastic interpreta- tion of the time evolution of nonlinear waves.

It is central importance that the formulation of the dynamics in terms of a master equation acting on a discretized phase space naturally leads to simulation algorithms which are well-suited for numerical (parallel) implemen- tation. Let us briefly sketch the main advantages of our method:

1. The numerical algorithms are easy to implement and very transparent. For example, the FORTRAN-code for the simulation of the shock wave solution of Burgers' equation consists of 120 statements.

2. The method of stochastic simulation is particularly well-suited for the in- tegration of dynamical systems with many degrees of freedom [2, 3]. This fact may be decisive for the study of fluid dynamical phenomena in higher dimensions.

3. Main parts of our simulation algorithms operate with arrays of integer variables thereby avoiding roundoff errors. For example, conservation laws which are known from the equations of motion can be fulfilled exactly.

4. The stochastic simulation algorithm was tested with the help of the shock wave solution of Burgers' equation at very large (R = 10 l~ Reynolds numbers: It turns out that within our stochastic approach numerical artefacts such as the Gibbs-phenomenon [11] or numerical viscosity [12] can easily be avoided.

The generalization to two and three space dimensions is straightforward but beyond the scope of the present work. The corresponding generalized master equations and their application will be published elsewhere. Concluding, we believe that both the conceptual transparency and the numerical simplicity and flexibility constitute a very powerful tool which might allow the treatment of real practical problems.

References

1. Honerkamp, J.: Stochastische Dynamische Systeme. VCH, Weinheim, 1990 2. Breuer, H. P.; Honerkamp, J.; Petruccione, F.: Chem. Phys. Lett. 190 (1992) 199 3. Breuer, H.P.; Honerkamp, J.; Petruccione, F.: Computational Polymer Science

1 (1991) 233 4. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-

Holland, Amsterdam. 1981 5. Burgers, J.M.: in Statistical Models and Turbulence, edited by Ehlers, J.;

Hepp, K.; Weidenmiiller, H.A. Lecture Notes in Physics 12, Springer-Verlag, Berlin, 1972 p. 41

A stochastic approach to computational fluid dynamics 267

6. Whitham, G. B. : Linear and Nonlinear Waves. John Wiley & Sons, New York, 1974 7. Landau, L.D. ; Lifshitz, E.M. : Fluid Mechanics, Volume 6 of Course of Theoretical

Physics. Pergamon Press, London, 1959 8. Breuer, H.P.; Petruccione, E: Hydrostochastics and Hydrodynamic Fluctuations,

preprint THEP 11/92, University of Freiburg (1992) 9. Sabelfeld, K. K.: Monte Carlo Methods in Boundary Value Problems. Springer,

Berlin 1991 10. Marshall, G.: Computer Physics Communications 56 (1989) 51 11. Canuto, C. ; Hussaini, M. Y.; Quarteroni, A. ; Zeng, T. A. : Spectral Methods in

Fluid Dynamics. Springer-Verlag, New York 1987. 12. Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer-Verlag

New York 1983

H. P. Breuer E Petruccione Alber t-Ludwigs-Universit~t Fakult~it for Physik Hermann-Herder-Str. 3 W-7800 Freiburg i. Br. FRG

Received June 12, 1992