Fluid Mechanics: Numerical Methods. In J.-P. Francoise, G.L. Naber and S.T. Tsou (eds.)Encyclopedia of Mathematical Physics vol. 2, pp. 365374 (2006). Oxford: Elsevier. (ISBN 978-0-1251-2666-3)

Jean-Luc Guermond, LIMSI, CNRS, University Paris

Sud, Orsay, France and Texas A&M University, Col-

lege Station, Texas

The objective of this article is to give an overviewof some advanced numerical methods commonlyused in fluid mechanics. The focus is set primarilyon finite elements methods and finite volume meth-ods.

1. Fluid Mechanics Models

Let be a domain in Rd (d = 2, 3) with bound-ary and outer unit normal n. is assumed tobe occupied by a fluid. The basic equations govern-ing fluid flows are derived from three conservationprinciples: conservation of mass, momentum, andenergy. Denoting the density by , the velocity byu , and the mass specific internal energy by ei, theseequations are

t+ (u) = 0(1)

t(u) + (u u) = + f ,(2)

t(ei) + (uei) = : + qT jT ,(3)

where is the stress tensor, = 12 (u + u)T is

the strain tensor, f is a body force per unit mass(gravity is a typical example), qT is a volume source(it may model chemical reactions, Joule effects, ra-dioactive decay, etc.), and jT is the heat flux. In ad-dition to the above three fundamental conservationequations, one may also have to add L equations

that accounts for the conservation of other quan-tities, say , 1 L. These quantities maybe the concentration of constituents in an alloy, theturbulent kinetic energy, the mass fractions of vari-ous chemical species by unit volume, etc. All theseconservation equations take the following form

(4) t()+(u) = qj , 1 L.

Henceforth the index is dropped to alleviate thenotation.

The above set of equations must be supple-mented with initial and boundary conditions. Typ-ical initial conditions are |t=0 = 0, u|t=0 = u0,and |t=0 = 0. Boundary conditions are usu-ally classified into two types: the essential bound-ary conditions and the natural boundary conditions.Natural conditions impose fluxes at the boundary.Typical examples are

(n + Ru)| = au,

(jT n + rT ei)| = aT

and

(jn + r)| = a.

The quantities R, rT , r, au, aT , a are given.Essential boundary conditions consist of enforcingboundary values on the dependent variables. Onetypical example is the so-called no-slip boundarycondition: u| = 0.

The above system of conservation laws is closedby adding three constitutive equations whose pur-pose is to relate each field , jT , j to the fields ,u, and . They account for microscopic propertiesof the fluid and thus must be frame independent.

366 Fluid Mechanics: Numerical Methods

Depending on the constitutive equations and ade-quate hypotheses on time and space scales, variousmodels are obtained. An important class of fluidmodel is one for which the stress tensor is a linearfunction of the strain tensor, yielding the so-calledNewtonian fluid model:

(5) = (p+ u)I + 2.

Here p is the pressure, I is the identity matrix, and and are viscosity coefficients. Still assuminglinearity, common models for heat and solute fluxesconsist of assuming

(6) jT = T, j = D,

where T is the temperature. These are the so-calledFouriers law and Ficks law, respectively.

Having introduced two new quantities, namelythe pressure p and the temperature T , two newscalar relations are needed to close the system.These are the state equations. One admissibleassumption consists of setting = (p, T ). An-other usual additional hypothesis consists of as-suming that the variations on the internal energyare proportional to those on the temperature, i.e.,ei = cPT .

Let us now simplify the above models by assum-ing that is constant. Then, mass conservation im-plies that the flow is incompressible, i.e., u = 0.Let us further assume that neither , , nor p de-pend on ei. Then, upon abusing the notation andstill denoting by p the ratio p/, the above setof assumptions yields the so-called incompressibleNavierStokes equations:

u = 0,(7)

tu + uu u + p = f .(8)

As a result, the mass and momentum conservationequations are independent of that of the energy andthose of the solutes:

cP (tT + uT ) (T ) = 2: + qT ,(9)

t+ u1(D) = 1

q.(10)

Another model allowing for a weak dependencyof on the temperature, while still enforcing incom-pressibility, consists of setting = 0(1(TT0)).If buoyancy effects induced by gravity are impor-tant, it is then possible to account for those by set-ting f = 0g(1 (T T0)), where g is the grav-ity acceleration, yielding the so-called Boussinesqmodel.

Variations on these themes are numerous and awide range of fluids can be modeled by using nonlin-ear constitutive laws and nonlinear state laws. Forthe purpose of numerical simulations, however, it isimportant to focus on simplified models.

2. The Building Blocks

From the above considerations we now extract asmall set of elementary problems which constitutethe building blocks of most numerical methods influid mechanics.

2.1. Elliptic equations. By taking the divergenceof the momentum equation (8) and assuming u tobe known and renaming p to , one obtains thePoisson equation

(11) = f,

where f is a given source term. This equation playsa key role in the computation of the pressure whensolving the NavierStokes equations; see (54b). As-suming adequate boundary conditions are enforced,this model equation is the prototype for the classof the so-called elliptic equations. A simple gen-eralization of the Poisson equation consists of theadvection-diffusion equation

(12) u() = f,

where > 0. Admissible boundary conditions are(n + r)| = a, r 0, or | = a. Thistype of equation is obtained by neglecting the timederivative in the heat equation (9) or in the soluteconservation equation (10). Mathematically speak-ing (12) is also elliptic since its properties (in par-ticular, the way the boundary conditions must beenforced) are controlled by the second-order deriva-tives. For the sake of simplicity, assume that u = 0in the above equation and that the boundary con-dition is | = 0, then it is possible to show that solves (12) if and only if minimizes the followingfunctional

J () =

(||2 f) dx,

where | | is the Euclidean norm and spans

(13) H = {;

||2 dx

Fluid Mechanics: Numerical Methods 367

for all H. This is the so-called variational for-mulation of (12). When u is not zero, no varia-tional principle holds but a similar way to refor-mulate (12) consists of multiplying the equation byarbitrary functions in H and integrating by partsthe second-order term to give

(14)

(u) + =

f, H.

This is the so-called weak formulation of (12). Weakand variational formulations are the starting pointfor finite element approximations.

2.2. Stokes equations. Another elementarybuilding block is deduced from (8) by assumingthat the time derivative and the nonlinear termare both small. The corresponding model is theso-called Stokes equations

u + p = f ,(15)

u = 0.(16)

Assume for the sake of simplicity that the no-slipboundary condition is enforced: u| = 0. Intro-duce the Lagrangian functional

L(v, q) =

(u:v qv f v) dx.

Set

X = {v;

|v|2 dx

368 Fluid Mechanics: Numerical Methods

Furthermore, (22) becomes

(24) dt[

(X(x, s; t), t)]

= f(X(x, s; t), t).

Then

(x, t) = 0(X(x, t; 0)) +

t

0

f(X(x, t; ), ) d

provided X(x, t; ) for all [0, t]. This showsthat the concept of characteristic curves is impor-tant to construct an approximation to (22).

3. Meshes

The starting point of every approximation tech-nique for solving any of the above model problemsconsists of defining a mesh of on which the ap-proximate solution is defined. To avoid having toaccount for curved boundaries, let us assume the do-main is a two-dimensional polygon (resp. three-dimensional polyhedron). A mesh of , say Th, is apartition of into small cells hereafter assumed tobe simple convex polygons in two dimensions (resp.polyhedrons in three dimensions), say triangles orquadrangles (resp. tetrahedrons or cuboids). More-over, this partition is usually assumed to be suchthat if two different cells have a nonempty intersec-tion, then the intersection is a vertex, or an entireedge, or an entire face. The left panel of Figure 1shows a mesh satisfying the above requirement. Themesh in the right panel is not admissible.

Fig 1: Admissible (left) and nonadmissible (right)

meshes.

4. Finite Elements: Interpolation

The finite element method is foremost an inter-polation technique. The goal of this section is toillustrate this idea by giving examples.

Let Th = {Km}1mNel be a mesh composed ofNel simplices, i.e., triangles in two dimensions ortetrahedrons in three dimensions. Consider the fol-lowing vector spaces of functions(25)Vh = {vh C

0(); vh|Km Pk, 1 m Nel},

where Pk denotes the space of polynomials of globaldegree at most k. Vh is called a finite element ap-proximation space. We now construct a basis forVh.

Given a simplex Km in Rd, let vn be a vertex

of Km, let Fn be the face of Km opposite to vn,and define nn to be the outward normal to Fn,1 n d+ 1. Define the barycentric coordinates

(26) n(x) = 1(x vn) nn(vl vn) nn

, 1 n d+ 1,

where vl is an arbitrary vertex in Fn (the definitionof n is clearly independent of vl provided vl be-longs to Fn). The barycentric coordinate n is anaffine function; it is equal to 1 at vn and vanisheson Fn; its level-sets are hyperplanes parallel to Fn.The barycenter of Km has barycentric coordinates

(1

d+ 1, . . . ,

1

d+ 1).

The barycentric coordinates satisfy the followingproperties: For all x Km, 0 n(x) 1, andfor all x Rd,

d+1

n=1

n(x) = 1, and

d+1

n=1

n(x)(x vn) = 0.

Consider the set of nodes {an,m}1nnsh of Kmwith barycentric coordinates(

i0k, . . . ,

idk

)

, 0 i0, . . . , id k, i0+. . .+id = k.

These points are called the Lagrange nodes of Km.It is clear that there are nsh =

12 (k + 1)(k + 2) of

these points in two dimensions and nsh =16 (k +

1)(k + 2)(k + 3) in three dimensions. It is remark-able that nsh = dim Pk.

Let {b1, . . . , bN} =

KmTh{a1,m, . . . ,ansh,m}

be the set of all the Lagrange nodes in the mesh.For Km Th and n {1, . . . , nsh}, let j(n,m) {1, . . . , N} be the integer such that an,m = bj(n,m);j(n,m) is the global index of the Lagrange nodean,m. Let {1, . . . , N} be the set of functions inVh defined by i(bj) = ij , then it can be shownthat

(27) {1, . . . , N} is a basis for Vh.

The functions i are called global shape functions.An important property of global shape functions isthat their supports are small sets of cells. More pre-cisely, let i {1, . . . N} and let Vi = {m; n; i =j(n,m)} be the set of cell indices to which the node

Fluid Mechanics: Numerical Methods 369

bi belongs, then the support of i is mViKm. Fork = 1, it is clear that i|Km = n for all m Vi andall n such that i = j(n,m), and i|Km = 0 other-wise. The graph of such a shape function in two di-mensions is shown in the left panel of Figure 2. Fork = 2, enumerate from 1 to d+1 the vertices of Km,and enumerate from d+2 to nsh the Lagrange nodeslocated at the midedges. For a midedge node of in-dex d+ 2 n nsh, let b(n), e(n) {1, . . . , d+ 1}be the two indices of the two Lagrange nodes atthe extremities of the edge in question. Then, therestriction to Km of a P2 shape function i is

(28) i|Km =

{

n(2n 1), if 1 n d+ 1,

4b(n)e(n), if d+ 2 n nsh,

Figure 2 shows the graph of two P2 shape functionsin two dimensions.

Fig 2: Two-dimensional Lagrange shape functions:

Piecewise P1 (left) and Piecewise P2 (center and right).

Once the space Vh is introduced, it is natural todefine the interpolation operator

(29) h : C0() v 7

N

i=1

v(bi)i Vh.

This operator is such that for all continuous func-tion v, the restriction of h(v) to each mesh cell isa polynomial in Pk and h(v) takes the same val-ues as v at the Lagrange nodes. Moreover, settingh = maxKmTh diam(Km), and defining

rLp = (

|r|p dx)1p for 1 p 0,

+ 12u 0, and r 0. Define

a(, ) =

((+ ) + ) dx

+

r ds.

Then, the weak formulation of (21) is: Seek H(H defined in (13)) such that for all H

(31) a(, ) =

f dx +

g ds.

Using the approximation space Vh defined in(25) together with the basis defined in (27), weseek an approximate solution to the above prob-

lem in the form h =N

i=1 Uii Vh. Then, asimple way of approximating (31) consists of seek-ing U = (U1, . . . , UN )

T RN such that for all1 i N

(32) a(h, i) =

fi dx +

gi ds.

This problem finally amounts to solving the follow-ing linear system

(33) AU = F,

where Aij = a(j , i) and Fi =

fi dx +

gi ds. The above approximation technique is

usually referred to as the Galerkin method. The fol-lowing error estimate can be proved

(34) hLp + h( h)Lp

chk+1Ck+1(),

370 Fluid Mechanics: Numerical Methods

where in addition to depending on the shape-regularity of the mesh, the constant c also dependson , , and .

5.2. Stokes equations. The line of thought devel-oped above can be used to approximate the NavierStokes problem (15)(16). Let us assume that thenonlinear term uu is linearized in the form vu,where v is known. Let Th be a mesh of , and as-sume that finite element approximation spaces havebeen constructed to approximate the velocity andthe pressure, say Xh and Mh. Assume for the sakeof simplicity that Xh X and Mh M . As-sume that bases for Xh and Mh are at hand, say{1, . . . ,Nu} and {1, . . . , Np}, respectively. Set

a(u,) =

((vu) + u: dx

and

b(v, ) =

v dx.

Then, we seek an approximate velocity uh =Nu

i=1 Uii and an approximate pressure ph =Np

k=1 Pkk such that for all i {1, . . . , Nu} andall k {1, . . . , Np} the following holds

a(uh,i) + b(i, ph) =

f i dx,(35)

b(uh, k) = 0.(36)

Define the matrix A RNu,Nu such that Aij =a(j ,i). Define the matrix B R

Np,Nu such thatBki = b(i, k). Then, the above problem can berecast into the following partitioned linear system

(37)

[

A BT

B 0

] [

UP

]

=

[

F0

]

,

where the vector F RNu is such that Fi =

f i.An important aspect of the above approximation

technique is that for the linear system to be invert-ible, the matrix BT must have full row rank (i.e., Bhas full column rank). This amounts to

(38) h > 0, infqhMh

supvhXh

qhvh dx

vhXqhM h,

where

vh2X =

|vh|2 dx, qh

2M =

q2h dx.

This nontrivial condition is called theLadyzenskajaBabuskaBrezzi condition (LBB) in

the literature. For instance, if P1 finite elementsare used to approximate both the velocity and thepressure, the above condition does not hold, sincethere are nonzero pressure fields qh in Mh suchthat

qhvh dx = 0 for all vh in Xh. Such fields

are called spurious pressure modes. An example isshown in Figure 3. The spurious function alterna-tively takes the values 1, 0, and +1 at the verticesof the mesh so that its mean value on each cell iszero.

0

0

00

0

0

0

0

00

00

1

1

1

1

1

1

1

1

1

1

1

1

+1+1

+1

+1

+1+1

+1

+1

+1+1

+1+1

Fig 3: The P1/P1 finite element: the mesh (left); one

pressure spurious mode (right).

Couples of finite element spaces satisfying theLBB condition are numerous. For instance, assum-ing k 2, using Pk finite elements to approximatethe velocity and Pk1 finite elements to approx-imate the pressure is acceptable. Likewise usingQk elements for the velocity and Qk1 elements forthe pressure on meshes composed of quadrangles orcuboids is admissible.

Approximation techniques for which the pressureand the velocity degrees of freedom are not associ-ated with the same nodes are usually called stag-gered approximations. Staggering pressure and ve-locity unknowns is common in solution methods forthe incompressible Stokes and NavierStokes equa-tions; see also 7.3.

6. Finite Volumes: Principles

The finite volume method is an approximationtechnique whose primary goal is to approximateconservation equations, whether time-dependent ornot. Given a mesh, say Th = {Km}1mNel , and aconservation equation

(39) t+ F (,,x, t) = f,

( = 0 if the problem is time-independent and = 1 otherwise), the main idea underlying everyfinite volume method is to represent the approx-imate solution by its mean values over the mesh

Fluid Mechanics: Numerical Methods 371

cells (K1 , . . . , KNel )T RNel and to test the con-

servation equation by the characteristic functionsof the mesh cells {1K1 , . . . , 1KNel }. For each cellKm Th, denote by nKm the outward unit nor-mal vector and denote by Fm the set of the facesof Km. The finite volume approximation to (39)consists of seeking (K1 , . . . , KNel )

T RNel such

that the function h =Nel

m=1 Km1K1m satisfiesthe following: For all 1 m Nel

(40) |Km|dtKm(t) +

Fm

Fm,h (h,hh, t) =

K

f dx,

where

|Km| =

K

dx,

hh is an approximation of , and Fm,h is an

approximation of

F (,,x, t)nKm d.

The precise definition of the so-called approximateflux Fm,h depends on the the nature of the problem(e.g., elliptic, parabolic, hyperbolic, saddle point)and the desired accuracy. In general the approxi-mate fluxes are required to satisfy the following twoimportant properties:1. Conservativity: For Km,Kl Th such that

= Km Kl, Fm,h = F

l,h .

2. Consistency: Let be the solution to (39), and

set h =1K1|K1|

K1 dx+. . .+

1KNel|KNel |

KNel dx, then

Fm,h (h,hh, t)h0

F (,,x, t)nd.

The quantity

|Fm,h (h,hh, t)

F (,,x, t)n d|

is called the consistency error.Note that (40) is a system of Ordinary Differ-

ential Equations (ODEs). This system is usuallydiscretized in time by using standard time-marchingtechniques such as explicit Euler, Runge-Kutta, etc.

The discretization technique above described issometimes referred to as cell-centered finite volumemethod. Another method, called vertex-centeredfinite volume, consists of using the characteristicfunctions associated with the vertices of the meshinstead of those associated with the cells.

7. Finite Volumes: Examples

In this section we illustrate the ideas introducedabove. Three examples are developed: the Poissonequation, the transport equation, and the Stokesequations.

7.1. Poisson problem. Consider the Poissonequation (11) equipped with the boundary condi-tion n| = a. To avoid technical details, assume

that =]0, 1[d. Let Kh be a mesh of composedof rectangles (or cuboids in three dimensions).

The flux function is F (,,x) = ; hence,Fm,h must be a consistent conservative approxima-tion of

nKm d. Let be an interior face of

the mesh and let Km,Kl be the two cells such that = Km Kl. Let xKm ,xKl be the barycentersof Km and Kl, respectively. Then, an admissibleformula for the approximate flux is

(41) Fm,h = ||

|xKm xKl |(Kl Km),

where || =

d. The consistency error is O(h)

in general, and is O(h2) if the mesh is composed ofidentical cuboids. The conservativity is evident. If is part of , an admissible formula for the ap-proximate flux is Fm,h =

ad. Then, upon

defining F iKm = FKm\ and FKm

= FKm ,the finite volume approximation of the Poissonproblem is: Seek h R

Nel such that for all1 m Nel

(42)

FiKm

Fm,h =

Km

f dx +

FKm

ad.

7.2. Transport equation. Consider the transportequation

t+ (u) = f,(43)

|t=0 = 0, | = a,(44)

where u(x, t) is a given field in C1( [0, T ]). LetTh be a mesh of . For the sake of simplicity, letus use the explicit Euler time-stepping to approxi-mate (40). Let N be positive integer, set t = T

N,

set tn = nt for 0 n N , and partition [0, T ] asfollows

[0, T ] =

N1

n=0

[tn, tn+1].

372 Fluid Mechanics: Numerical Methods

Denote by nh RNel the finite volume approxi-

mation of h(tn). Then, (40) is approximated as

follows|Km|t (

n+1Km

nKm) +

Fm

Fm,h (h,hh, tn) =

K

f(x, tn) dx,(45)

where 0Km =

Km0 dx. The approximate flux

Fm,h must be a consistent conservative approxi-mation of

(unKm)d. Let be a face of the

mesh and let Km, Kl be the two cells such that = Km Kl (note that if is on , belongs toone cell only and we set Km = Kl). If is on

,set

(46) Fm,h =

(unKm)ad.

If is not on , set unm, =

(unKm) d and

define

(47) Fm,h =

{

nKmunm, if u

nm, 0,

nKlunm, if u

nm, < 0.

The above choice for the approximate flux is usu-ally called the upwind flux. It is consistent with theanalysis that has been done for (22), i.e., informa-tion flows along the characteristic lines of the fieldu; see (24). In other words, the updating of n+1kmmust be done by using the approximate values nhcoming from the cells that are upstream the flowfield.

An important feature of the above approxima-tion technique is that it is L-stable in the sensethat

max0nN,1mNel

|nKm | c(u0, f)

if the two mesh parameters t and h satisfy theso-called CourantFriedrichsLevy (CFL) conditionuLt/h c(), where c() is a constant thatdepends on the mesh regularity parameter =maxKmTh hKm/Km . In one dimension c() = 1.

7.3. Stokes equations. To finish this short reviewof finite volume methods, we turn our attention tothe Stokes problem (15)(16) equipped with the ho-mogeneous Dirichlet boundary condition u| = 0.

Let Th be a mesh of composed of triangles (ortetrahedrons). All the angles in the triangulationare assumed to be acute so that for all K Th, theintersection of the orthogonal bisectors of the sidesof K, say xK , is in K. We propose a finite volume

approximation for the velocity and a finite elementapproximation for the pressure. Let {e1, . . . ,ed} bea Cartesian basis for Rd. Set 1kKm = 1Kmek for all1 m Nel and 1 k d, then define

Xh = span{11K1, . . . ,1dK1 , . . . ,1

1KNel

, . . . ,1dKNel}.

Let {b1, . . . , bNv} be the vertices of the mesh, andlet {1, . . . , Nv} be the associated piecewise linearglobal shape functions. Then, set

Nh = span{1, . . . , Nv},

Mh = {q Nh;

q dx = 0};

see 4. The approximate problem consists of seek-ing (uK1 , . . . ,uKNel ) R

dNel and ph Mh suchthat for all 1 m Nel, 1 k d, and all1 i Nv,

Fm

1kKmFm,h + c(1

kKm

, ph) =

Km

1kKmf dx,(48)

c(uKm , i) = 0,(49)

where

c(vKm , ph) =

Km

vKm ph dx.

Moreover,

Fm,h =

||

|xm xl|(uKm uKl) if = Km Kl,

||

d(xm, )uKm if = Km ,

where d(xKm , ) is the Euclidean distance betweenxKm and . This formulation yields a linear sys-tem with the same structure as in (37). Note inparticular that

(50) supvhXh

c(vh, ph)

vhL= phL1 ,

Since the mean-value of ph is zero, phL1 is anorm on Mh. As a result, an inequality similar to(38) holds. This inequality is a key step to provingthat the linear system is wellposed and the approx-imate solution converges to the exact solution of(15)(16).

Fluid Mechanics: Numerical Methods 373

8. Projection methods for NavierStokes

In this section we focus on the time approxima-tion of the NavierStokes problem:

tu u + uu + p = f ,(51a)

u = 0,(51b)

u| = 0,(51c)

u|t=0 = u0.(51d)

where f is a body force and u0 is a solenoidal ve-locity field. There are numerous ways to discretizethis problem in time, but, undoubtedly, one of themost popular strategies is to use projection meth-ods, sometimes also referred to as ChorinTemammethods.

A projection method is a fractional-step time-marching technique. It is a predictorcorrectorstrategy aiming at uncoupling viscous diffusion andincompressibility effects. One time step is com-posed of three substeps: in the first substep, thepressure is made explicit and a provisional veloc-ity field is computed using the momentum equa-tion; in the second substep, the provisional velocityfield is projected onto the space of incompressible(solenoidal) vector fields; in the third substep, thepressure is updated.

Let q > 0 be an integer and approximate the timederivative of u using a backward difference formulaof order q. To this end, introduce a positive integerN , set t = T

N, set tn = nt for 0 n N , and

consider a partitioning of the time interval in theform

[0, T ] =N1

n=0

[tn, tn+1].

For all sequences vt = (v0,v1, . . . ,vN ), set

(52) D(q)vn+1 = qvn+1

q1

j=0

jvnj ,

where q 1 n N 1. The coefficients j aresuch that

1

t(qu(t

n+1)

q1

j=0

ju(tnj))

is a qth-order backward difference formula approx-imating tu(t

n+1). For instance,

D(1)vn+1 = vn+1 vn,

D(2)vn+1 = 32vn+1 2vn + 12v

n1.

Furthermore, for all sequences t =(0, 1, . . . , N ), define

(53) ,n+1 =

q1

j=0

jnj ,

so thatq1

j=0 jp(tnj) is a (q1)th-order extrapo-

lation of p(tn+1). For instance, p,n+1 = 0 for q = 1,p,n+1 = pn for q = 2, and p,n+1 = 2pn pn1 forq = 3. Finally, denote by (uu),n+1 a q-th orderextrapolation of (uu)(tn+1). For instance,

(uu),n+1 =

{

unun if q = 1,

2ununun1un1 if q = 2.

A general projection algorithm is as follows. Setu0 = u0 and

l = 0 for 0 l q 1. If q > 1,assume that u1, . . . , uq1, p,q and (uu),q havebeen initialized properly. For n q 1, seek un+1

such that un+1| = 0 and

(54a) D(q)

t un+1 un+1

+ (

p,n+1 +q1

j=0jt

nj)

= Sn+1,

where Sn+1 = f(tn+1) (uu),n+1. Then solve

(54b) n+1 = un+1, nn+1| = 0.

And finally update the pressure as follows:

(54c) pn+1 =qt

n+1 + p,n+1 un+1.

The algorithm (54a)(54b)(54c) is known in theliterature as the rotational form of the pressure-correction method. Upon denoting ut =(u(t0), . . . ,u(tN )) and pt = (p(t

0), . . . , p(tN )), theabove algorithm has been proved to yield the follow-ing error estimates

ut ut2(L2) ct2,

(ut ut)2(L2) + pt pt2(L2) ct32 .

where t22(L2) = t

Nn=0

|n|2 dx.

A simple strategy to initialize the algorithm con-sists of using D(1)u1 at the first step in (54a), thenusing D(2)u2 at the second step, and proceedinglikewise until u1, . . . , uq1 have all been computed.

At the present time, projection methods countamong the few methods that are capable of solvingthe time-dependent incompressible NavierStokesequations in three dimensions on fine meshes withinreasonable computation times. The reason for this

374 Fluid Mechanics: Numerical Methods

success is that the unsplit strategy, which consistsof solving

D(q)

t un+1 un+1 + pn+1 = Sn+1,(55a)

un+1 = 0; un+1| = 0,(55b)

yields a linear system similar to (37), which usu-ally takes far more time to solve than sequentiallysolving (54a) and (54b). It is commonly reportedin the literature that the ratio of the CPU time forsolving (55a)(55b) to that for solving (54a)(54b)(54c) ranges between 10 to 30.

See Also

Fluid Dynamics. Incompressible Euler equa-tions. Newtonian fluids and thermohydraulics.Navier-Stokes turbulence. Partial differential equa-tions in fluid mechanics. Variational methods inturbulence.

Further Reading

Doering, C.R.; Gibbon, J.D. (1995) Appliedanalysis of the Navier-Stokes equations. CambridgeTexts in Applied Mathematics. Cambridge: Cam-bridge University Press.

Ern A. and Guermond J.-L. (2004) Theory andPractice of Finite Elements. Springer Series in Ap-plied Mathematical Sciences, Vol. 159. New-York:Springer-Verlag.

Eymard R., Gallouet T., and Herbin, R. (2000)Finite volume methods. In: Ciarlet PG and LionsJL (eds.) Handbook of numerical analysis, Vol. VII7131020. Amsterdam: North-Holland.

Karniadakis, G.E.; Sherwin, S.J. (1999)Spectral/hp element methods for CFD. Numeri-cal Mathematics and Scientific Computation. NewYork: Oxford University Press.

Rappaz M., Bellet M. and Deville M. (2003) Nu-merical Modeling in Material Science and Engineer-ing. Springer Series in Computational Mathemat-ics, Vol. 32. Berlin: Springer-Verlag.

Temam, R. (1984) Navier-Stokes equations.Theory and numerical analysis. Studies in Math-ematics and its Applications, Vol. 2. Amsterdam:North-Holland.

Toro, E.F. (1997) Riemann solvers and numeri-cal methods for fluid dynamics. A practical intro-duction. Berlin: Springer.

Wesseling, P. (2001) Principles of computationalfluid dynamics. Springer Series in ComputationalMathematics, Vol. 29. Berlin: Springer.