The Control-Volume Finite-Difference

Approximation to the Diffusion Equation

Gerald Recktenwald∗

January 28, 2019

Abstract

A two-dimensional computer code for solution of the diffusion equation (Poisson equation)is described. The code is implemented in Matlab, and is intended for educational use. Thepartial differential equation is converted to a system of linear equations with the finite-volumemethod. The system is solved by a direct method, though extending the code to use iterativemethods would not be difficult. Use of the code requires the user to edit and run Matlabfunctions. Utility routines for prescribing a variety of boundary conditions and computing wallfluxes are provided to ease the programming tasks. The code is demonstrated with a handfulof sample problems.

1 Introduction

The control-volume finite-difference (CVFD) method is widely used in the numerical simulation offluid dynamics, heat transfer and combustion. Several commercial CFD (computational fluid dy-namics) codes are based on this method. Though there is no one ultimate numerical approximationscheme, the CVFD method has numerous desirable features. It naturally maintains conservation ofspecies when applied to conservation laws. It readily handles material discontinuities and conjugateheat transfer problems. It also has the pedagogical advantage that only simple calculus is requiredto derive the CVFD approximation to common conservation equations.

This article presents a brief derivation of the CVFD approximation to the diffusion equation.The two dimensional diffusion equation in Cartesian coordinates is

∂

∂x

(Γ∂φ

∂x

)+∂

∂y

(Γ∂φ

∂y

)+ S = 0 (1)

where φ is the scalar field, Γ is the diffusion coefficient, and S is the source term. In this form Γis assumed to be non-uniform. We seek the solution φ(x, y) to Equation (1) in a region of space Ω,subject to appropriate conditions on the boundary, ∂Ω.

The article begins with the nomenclature necessary to define a finite volume grid on a rectangulardomain. Following that the general form of the CVFD approximation to the diffusion equation isobtained. Next, the harmonic mean is introduced as the correct procedure for treating discontinuitiesin Γ. In § 5 important details concerned with writing a CVFD program are discussed. A series ofmodel problems suitable for testing a CVFD code problems are presented in § 6. The article endswith a discussion of truncation error, including a quantitative procedure for verifying the correctnessof a CVFD code.

∗Mechanical and Materials Engineering Department, Portland State University, Portland, OR, 97201,gerry@pdx.edu

2 FINITE VOLUME MESH 2

x

y

1 2 nx

i = 0 2j = 0

2

3

3 nx

ny

xu(1) = 0

x(i)

y(j)

xu(i)

1

yv(j)

yv(ny+1)

xu(nx+1)

1

nx+1 nx+2

nx ny

yv(1) = 0

Interior node

Boundary node

Ambiguous corner node

Figure 1: The control-volume finite-difference grid covering the calculation domain.

2 Finite Volume Mesh

Figure 1 depicts a rectangular domain of length Lx in the x-direction and Ly in the y-direction.The domain is divided into non-overlapping control volumes by the lines that define the boundariesof the individual control volumes. The pattern created by the lines is called the computational gridor mesh. In Figure 1 the control volumes are square, but this is not required. At the center of eachcontrol volume is a node, designated by an open circle. Two sets of grid lines can be identified:the grid lines that define the control volume faces, and the grid lines (not shown) that define thelocations of the nodes. In the computer implementation the grid lines that define node locations willbe stored in the variables x(i) and y(j). The grid lines that define the control volume interfaceswill be stored in xu(i) and yv(j).

In section 3 we obtain an approximation to Equation (1) which applies to the interior nodes.Boundary values are applied at the nodes indicated by full circles. Nodes at the very corner ofthe domain, those marked with open squares, will not appear in the discrete approximation toEquation (1).

Nodes in the domain may be identified by their (i, j) grid indices. An additional naming con-vention based on the directions on a map will be used to simplify the algebra in following sections.Figure 2 is a detailed sketch of one of the control volumes in the domain. A typical node (i, j) inFigure 1 is also referred to as node P in Figure 2. The (i + 1, j) and (i − 1, j) neighbors of P aredesignated E for east and W for west, respectively. The (i, j + 1) and (i, j − 1) neighbors of P arereferred to as N and S for north and south, respectively. Figure 2 also defines numerous geometricvariables. In general the width, ∆x, of a control volume will not be equal to the distances δxe andδxw between P and its east and west neighbors. Regardless of the grid spacing P is always located

3 DISCRETE APPROXIMATION FOR INTERIOR CONTROL VOLUMES 3

δxw δx

e

S

PW E

N

∆x

∆y

xexw

yn

ys

δys

δyn

Figure 2: Geometric variables for a typical control-volume.

in the geometric center of its control volume. Thus

xP − xw = xe − xP =∆x

2(2)

yP − ys = yn − yP =∆y

2(3)

In these expressions it is crucial to distinguish between upper and lower case letters used as sub-scripts. Lower case subscripts refer to the locations of the control volume faces. Upper case sub-scripts refer to the locations of the nodes.

3 Discrete Approximation for Interior Control Volumes

The control volume finite-difference method is used to transform Equation (1) to a system of discreteequations for the nodal values of φ. First, Equation (1) is integrated over the typical control volumedepicted in Figure 2. This reduces the equation to one involving only first derivatives in space.Then these first derivatives are replaced with central difference approximations.

Integrating the first term in Equation (1) over the finite control volume yields∫ yn

ys

∫ xe

xw

∂

∂x

(Γ∂φ

∂x

)dx dy =

∫ yn

ys

[(Γ∂φ

∂x

)e

−(

Γ∂φ

∂x

)w

]dy (4)

≈[(

Γ∂φ

∂x

)e

−(

Γ∂φ

∂x

)w

]∆y (5)

≈[ΓeφE − φPδxe

− ΓwφP − φWδxw

]∆y (6)

Equation (4) follows from an exact evaluation of the inner integral with respect to x. Equation (5)is obtained by assuming that the flux terms, e.g. (Γ∂φ/∂x)e, do not vary in the y direction overa given control volume face. Equation (6) results from applying a central difference approximationto each partial derivative term. In these expressions φE , φW , and φP are the nodal values of φ,

4 NON-UNIFORM Γ 4

and Γe and Γw are the diffusion coefficients evaluated at the interfaces of the control volumes. Theprocedure for evaluating Γe and Γw is described in section 4.

Starting with the second term in Equation (1) and applying the steps used to obtain Equation (6)results in ∫ xe

xw

∫ yn

ys

∂

∂y

(Γ∂φ

∂y

)dy dx ≈

[ΓnφN − φPδyn

− ΓsφP − φSδys

]∆x (7)

On the left side of Equation (7) the order of integration has been chosen so that the outer ∂/∂yoperator is eliminated first. This is allowable if the integrand is smooth, as is the case for solutionsto Equation (1).

Integrating the source term in Equation (1) over the control volume gives∫ xe

xw

∫ yn

ys

S dy dx ≈ SP ∆x∆y (8)

where Sp is the source term evaluated at node P . Adding together Equations (6)–8 and rearrangingyields

−aSφS − aWφW + aPφP − aEφE − aNφN = b (9)

where

aE =Γe

∆x δxe(10)

aW =Γw

∆x δxw(11)

aN =Γn

∆y δyn(12)

aS =Γs

∆y δys(13)

aP = aE + aW + aN + aS (14)

b = SP (15)

For additional information on the transformation of Equation (1) into Equation (9) (see Patankar [3]).Equations 9 through 15 apply to each of the N = nx ny internal nodes in Figure 1. This defines alinear system of N equations in the N unknown internal values of φ.

4 Non-uniform Γ

Calculation of the coefficients in equations 10–13 requires values of Γ at the interfaces of the controlvolumes. Rather than using an interpolation scheme, the interface values of Γ are defined byrequiring continuity of the diffusive flux at the control volume interfaces. Consider the situationdepicted in Figure 3 where two control volumes meet along a material discontinuity. This commonsituation occurs when the diffusion equation is solved in a composite material.

Assume that Γ is uniform within each of the control volumes. (When the finite volume grid isdefined the control volume faces are aligned with discontinuities. Thus it is not an unreasonablerestriction to require that material properties are uniform within a control volume.) Continuity offlux at the interface requires

ΓP∂φ

∂x

∣∣∣∣xe−

= ΓE∂φ

∂x

∣∣∣∣xe+

= Γe∂φ

∂x

∣∣∣∣xe

(16)

4 NON-UNIFORM Γ 5

δxe

material 1 material 2

δxe+

δxe­

P E

Figure 3: Discontinuous variation in Γ at the interface between two control-volumes.

Equation (16) defines Γe. In other words, the value of Γe is chosen such that the flux, Γ∂φ/∂x, iscontinuous. Central difference approximations to the flux continuity conditions in Equation (16)are

ΓeφE − φPδxe

= ΓPφe − φPδxe−

(17)

ΓeφE − φPδxe

= ΓEφE − φeδxe+

(18)

where φe is the value of φ at the interface. Equations 17 and 18 are discrete approximations for thediffusion flux into the left (minus) and right (plus) sides of the control volume interface in Figure 3.Equations 17 and 18 can be rearranged as

φe − φP =δxe−ΓP

Γe

δxe(φE − φP ) (19)

φE − φe =δxe+ΓE

Γe

δxe(φE − φP ) (20)

Adding equations 19 and 20 gives

φE − φP =Γe

δxe(φE − φP )

[δxe−ΓP

+δxe+ΓE

].

Cancel the factor of (φE − φP ) and solve for Γe/δxe to get

Γe

δxe=

[δxe−ΓP

+δxe+ΓE

]−1=

ΓE ΓP

δxe−ΓE + δxe+ΓP.

Thus,

Γe =ΓE ΓP

βΓE + (1− β)ΓP(21)

5 IMPLEMENTATION 6

where

β ≡ δxe−δxe

=xe − xPxE − xP

(22)

Equation (21) gives the interface diffusion coefficient as the harmonic mean of the diffusion coeffi-cients in adjacent control volumes. This gives the proper limiting behavior as either ΓE or ΓP goesto zero. Similar formulas for Γw, Γn, and Γs may be derived, or written down by direct analogy tothe formula for Γe. These interface values for Γ are used in equations 10–13.

5 Implementation

A “good” code to solve Equation (1) will have efficient data structures, a robust and efficientsolution procedure, it will be flexible enough to accommodate practical problems, and it will beeasy to maintain and debug. There are many possible ways to implement a code to achieve theaforementioned goals. In this section, the design of a particular set of Matlab routines for thesolution of Equation (1) is described.

Table 1: Core Matlab routines used to solve Equation (1).

Function Description

fvAmatrix Store the finite volume coefficients aE , aW , aN , aS , and aPin the sparse matrix A in Equation (24).

fvave Compute the volume-weighted average of a field variableover the domain.

fvbc Modify finite volume coefficients and source terms to in-clude the effect of boundary conditions. The final value ofaP is also computed.

fvcoef Compute the neighboring finite volume coefficients (aE ,aW , aN , and aS) for all interior control volumes. No specialtreatment is applied to control volumes adjacent to bound-aries.

fvpost Update unknown boundary values and heat fluxes after thesolution to the interior nodes is obtained. An energy bal-ance is computed and printed. The temperature field isstored in a matrix suitable for use with Matlab contourand surface plotting routines.

fvUniformMesh Compute the location of cell centers and cell interfaces fora group of cells with uniform size. fvUniformMesh can beapplied to either x or y direction cells, and with care it canbe applied to subregions of the grid.

fvUniBlockMesh Compute the location of cell centers and cell interfaces fora mesh consisting of blocks that contain uniform meshes.The overall mesh can contain blocks of unequal size andunequal numbers of control volumes. Within each block,the mesh is uniform.

5 IMPLEMENTATION 7

The code described here was designed with three primary objectives.

• Make the code modular so that it can be more easily maintained.

• Make the code flexible so that it can be applied to a number of different problems describedby Equation (1). In particular, allow users of the code to easily select different boundaryconditions, material properties, and source terms.

• Make the code reasonably simple to understand. If necessary, sacrifice some efficiency orflexibility in order to make the code easier to read.

The solution to a particular problem is obtained by defining problem-specific grid and boundaryvalues, and then using a set of generic Matlab functions. The generic Matlab functions are listedin Table 1. Table 2 represents the structure of an analysis code that uses the functions from Table 1.The left column lists the sequence of steps in the main program. On the right shows the names ofthe Matlab routines that perform each step.

5.1 Mesh Variables

The mesh is specified by four vectors: x, xu, y, and yv. As shown in Figure 1, x(i) and y(j) arethe x and y positions of the node for φi,j . The indexing of the control volume faces is arbitrary. Wedefine xu(i) to be the location of the face to the left of x(i), and define yv(j) to be the location ofthe face below y(j). This convention comes from related codes that are used to model convectivetransport, which will not be discussed here.

For many problems either a uniform or block-uniform mesh are suitable. We define a uniformmesh to have uniform control volume widths in any one coordinate direction. The widths of thecontrol volumes in different directions need not be uniform. This situation is depicted in the lefthand side of Figure 4 where ∆x 6= ∆y, but ∆x is same for all control volumes. For a uniform mesh,the x, xu, y, and yv vectors are computed in fvUniformMesh.

A block-uniform mesh is shown in the right hand side of Figure 4. In a block-uniform mesh, thecalculation domain is divided into a number of panels in the x and y direction. The intersectionsof these panels define rectangular blocks. The width ∆x of control volumes is uniform within anx-direction panel, and adjacent panels can have different ∆x. Similarly, ∆y is uniform within ay-direction panel, but may vary from panel to panel. The fvUniBlockMesh function is used todefine block-uniform meshes.

Table 2: Relationship between the core Matlab functions and the main program used to solve aparticular problem.

Algorithm Tasks Core RoutinesDefine the mesh fvUniformMesh

Define boundary conditionsCompute finite-volume coefficients for interior cells fvcoef

Adjust coefficients for boundary conditions fvbc

Solve system of equationsAssemble coefficient matrix fvAmatrix

SolveCompute boundary values and/or fluxes fvpost

Plot results

5 IMPLEMENTATION 8

Uniform Mesh Block-Uniform Mesh

Lx1, nx1 Lx2, nx2

Ly1, ny1

Ly2, ny2

Ly3, ny3

x

y

∆x

∆y

Figure 4: Uniform and block-uniform meshes. Note that the uniform mesh the widths of the controlvolumes in the x and y directions are not necessarily uniform. The term uniform describes thespacing of the control volumes along individual directions.

For convenience, we define the auxiliary vectors dx, dy, dxw, dys

dx(i) = xu(i + 1)− xu(i)

dy(j) = yv(j + 1)− yv(j)

dxw(i) = x(i)− x(i− 1)

dys(j) = y(j)− y(y− 1)

dx(i) and dy(j) are the x- and y-direction widths of the control volume around the node for φi,j .These code variables correspond to ∆x and ∆y in Figure 2. dxw and dxs are the distances betweenadjacent nodes, which correspond to δx and δy in Figure 2.

5.2 Data Structures

The fundamental decision is whether the unknowns at the center of the control volume will bestored in one-dimensional arrays (vectors) or multidimensional arrays. Storing the data in vectorshas the potential for greater computational efficiency, whereas multidimensional arrays allow themost straightforward programming.

The grid in Figure 1 suggests that the discrete variable, φ, could be naturally stored as a two-dimensional array, phi(i,j). If the indices of phi(i,j) run 0 ≤ i ≤ nx + 1 and 0 ≤ j ≤ ny + 1then both the unknown interior values and the boundary conditions can be stored in the same array.The primary advantage of this data structure is that it is easy to program. Use of multidimensionalarrays does not preclude efficient programs. However, vector processors are more efficient whenoperating on longer vectors than shorter vectors. With φ stored in a two-dimensional array thelargest vector length is nx (if the code is written in Fortran) or ny (if the code is written in C).

Alternatively one can store the nodal values of φ at the interior grid points in a vector phi[n],n = 1,...,N. Data on the two-dimensional grid of Figure 1 can be mapped to the one-dimensionalarray with the so-called natural ordering

n = i+ (j − 1)nx

Other orderings are possible (see, e.g., [1, 2]).With natural ordering the neighbors in the compass point notation of Figure 2 have these indices

np = i + (j-1)*nx

5 IMPLEMENTATION 9

ne = np + 1

nw = np - 1

nn = np + nx

ns = np - nx

where np is the index of node (i, j), ne is the index of node (i + 1, j), etc. Whether or not theunknown values of φ are stored in a two-dimensional array corresponding to the grid indices or avector with natural ordering, the set of unknowns can be represented symbolically as a vector.

φ1,1φ2,1

...φnx,1φ1,2φ2,2

...φi,j

...φnx,ny

⇐⇒

φ1φ2...

φnxφnx+1

φnx+2

...φn...φN

(23)

If the interior values of φ are stored in a vector using natural ordering, additional storage for theboundary values must also be provided. One solution to this problem is to define four additionalvectors

ebcval(j), j = 1,..,ny

wbcval(j), j = 1,..,ny

nbcval(i), i = 1,..,nx

sbcval(i), i = 1,..,nx

The values stored in ebcval, etc., are the values of φ of the boundary nodes. This is sufficient forDirichlet boundary conditions. Additional information needs to be provided if multiple types ofboundary conditions (e.g., Neumann, symmetry, convective) are allowed in the code.

To allow for additional boundary types, the boundary condition information is stored in an n×5matrix for each of the four boundaries. The value of n is either nx or ny depending on the boundary:east and west boundaries are adjacent to ny control volumes, and the north and south boundariesare adjacent to nx control volumes. Table 3 documents the four types of boundary conditions thatare defined for the finite volume codes listed in Table 1.

The names and dimensions of the boundary condition matrices are listed in Table 4. The firstletter of the matrix data structure indicates the location of the boundary. Thus, ebc contains datafor the east boundary, wbc contains data for the west boundary, etc. The structure of these matricesis the same for each boundary surface.

Table 5 describes the contents of each column of ebc. The contents of the other boundarymatrices are analogous, except that the row index in nbc and sbc is the index of the boundary nodein the x direction, e.g. sbc(3,1) is the boundary condition type for the third cell from the left onthe south boundary.

5.3 Solving the Linear System

Equation (9) applies to each of the interior values of φ. Determining the discrete φ field requiressolving the system

Ax = b (24)

5 IMPLEMENTATION 10

Table 3: Boundary condition types. The boundary type is expressed in terms of a heat conductionproblem where φ = T , Γ = k, and S is a heat source term.

Boundarytype

BoundaryCondition Post-processing in fvpost

1 Specified T Compute q′′ from discrete approximation to Fourier’s law.

q′′ = kTb − Tixb − xi

where Ti and Tb are interior and boundary temperatures,respectively.

2 Specified q′′ Compute Tb from discrete approximation to Fourier’s law.

Tb = Ti + q′′xb − xik

where Ti and Tb are interior and boundary temperatures,respectively.

3 Convection From specified h and T∞, compute boundary temperatureand heat flux through the cell face on the boundary. Con-tinuity of heat flux requires

−kTb − Tixb − xi

= h(Tb − Tamb)

which can be solved for Tb to give

Tb =hTamb + (k/δxe)Ti

h+ (k/δxe)

where δxe = xb − xi4 Symmetry q′′ = 0. Set boundary Tb equal to adjacent interior Ti.

Table 4: Matrices for storing boundary condition data.

Matrix boundary size

ebc east ny× 5

wbc west ny× 5

nbc north nx× 5

sbc south nx× 5

5 IMPLEMENTATION 11

Table 5: Contents of ebc data structure for the east boundary. j is the index of the boundary nodein the y-direction.

Column ofebc Value

ebc(j,1) Integer from 1 to 4 indicating the type of boundary condition

ebc(j,2) Value of T on the boundary node (if known)

ebc(j,3) Value of heat flux through the cell face on the boundary (if known)

ebc(j,4) Value of convection coefficient h if ebc(j, 1) = 3.

ebc(j,5) Value of free-stream temperature of fluid flowing over the the bound-ary if ebc(j, 1) = 3.

A = a S apa W a E a N

Figure 5: Schematic representation of the A matrix.

where the unknown vector x corresponds to the interior values of φ (cf. Equation (23)), the coefficientmatrix, A, consists of the coefficients of Equation (9), and the right hand side vector b, includes thesource term (Equation (15)) and boundary conditions. The matrix A is sparse in the sense thatthere are many more zero entries than non-zero entries. The structure of the A matrix is depicted

6 MODEL PROBLEMS 12

in Figure 5. The b vector is

b =

f1,1 + aS1,1 φ1,0 + aW1,1 φ0,1f2,1 + aS2,1 φ2,0...fnx,1 + aSnx,1 φnx,0 + aEnx,1 φnx+1,1

f1,2 + aW1,2 φ0,2f2,2...fi,j...fnx,ny + aNnx,ny φnx,ny+1 + aEnx,ny φnx+1,ny

(25)

where the superscripts refer to the compass point notation for the coefficients in equations 10–13 andthe subscripts refer to the grid indices in Figure 1. In Equation (25) the φ values on the boundaryare referred to in terms of their two-dimensional grid indices. Storing φ as a vector is necessaryif the system of equations represented by 9 is to be solved with a direct technique, such as Gausselimination. For iterative methods the matrix A is never explicitly assembled. Rather the entriesin A are stored in arrays corresponding directly to the coefficients in Equation (9). Thus, iterativemethods can use the coefficient arrays whether they are stored as vectors or multidimensional arrays.

6 Model Problems

Computer codes to solve Equation (1) may be exercised by applying them to a series of modelproblems. These problems contain features found in more complicated engineering situations. Sincethe geometry and boundary conditions are relatively simple, the model problems may be solved withsimple codes that are not as complex as those capable of solving practical engineering problems.These model problems are also useful for debugging more complex codes and testing different solutionstrategies.

The model problems require solution of Equation (1) on a rectangular domain

0 ≤ x ≤ Lx 0 ≤ y ≤ Ly

The problems are distinguished by their different boundary conditions, and by the variation of thesource term and diffusivity in the domain.

Model Problem 1

Model problem 1 has φ = 0 on all boundaries, Γ = 1, and the source term given by

S =

[(π

Lx

)2+

(2π

Ly

)2]sin

(πx

Lx

)sin

(2πy

Ly

)(26)

The exact solution is

φ = sin

(πx

Lx

)sin

(2πy

Ly

)(27)

and is shown in Figure 6.

6 MODEL PROBLEMS 13

0

0.5

1

00.2

0.40.6

0.81

0

0.5

1

1.5

2

yx

0

0.5

1

00.2

0.40.6

0.81

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

yx

Figure 6: Solution to model problem 1 (left) and model problem 2 (right) on a 64× 64 mesh.

Model Problem 2

Model problem 2 has φ = 0 on all boundaries, Γ = 1, and a uniform source term, i.e. S(x, y) = con-stant. The solution is shown in the right hand side of Figure 6.

Model Problem 3

Model problem 3 involves solution to Equation (1) in the domain shown in Figure 7. The sourceterm and diffusion coefficient have different values in two subregions of the domain. Outside of thecentral region

Γ = Γ1 = 1 S = 0

In the central region of the domain

Γ = κΓ1 S = 1000

where κ > 0 is a scalar. When κ 6= 1 there is a discontinuous diffusivity change between thetwo regions. Model problem 3 becomes more difficult to solve as the magnitude of κ increases ordecreases. The boundary conditions are

φ(0, y) = 20 φ(Lx, y) = 0φ(x, 0) = 10 φ(x, Ly) = 0

Model Problem 4: Fully-Developed Flow in a Rectangular Duct

Figure 8 shows two representations of the cross section of a rectangular duct. For simple fully-developed flow the governing equation for the axial velocity w is

µ

[∂2w

∂x2+∂2w

∂y2

]− dp

dz= 0 (28)

The code for solving Equation (1) can be used to solve Equation (28) by making the followingdefinitions

φ = w, Γ = µ (uniform), S = −dpdz.

6 MODEL PROBLEMS 14

0.25Lx

0.25Ly

0.5Ly

0.5Lx

Lx

Ly

Γ1 = 1

S1 = 0

Γ2 = κΓ1

S2 = 1000

0

0.5

1

00.2

0.40.6

0.81

0

5

10

15

20

25

30

35

yx

Figure 7: Geometry of calculation domain (left) and finite-volume solution on a 64 × 64 mesh formodel problem 3.

For the full duct simulation depicted on the left hand side of Figure 8, the boundary conditionsare no slip conditions on all four walls.

w(x, 0) = w(x, Ly) = w(0, y) = w(Lx, y) = 0. (full duct)

For the quarter duct simulation depicted on the right hand side of Figure 8, the boundary conditionsare no slip conditions on the solid walls (x = Lx and y = Ly)

w(Ly, y) = w(x, Lx) = 0 (quarter duct)

and symmetry conditions on the other two planes

∂u

∂x

∣∣∣∣x=0

=∂u

∂y

∣∣∣∣y=0

= 0.

After the solution to the w field is obtained, the engineering quantity of interest is the product

Full Duct

x

y

Lx

Ly

Quarter Duct

x

y

Lx

Ly

Figure 8: Two possible calculation domains (shaded regions) for fully-developed flow in a rectangularduct.

7 TRUNCATION ERRORS 15

of the Darcy friction factor f , and the Reynolds number Re, where

f =Dh

12ρw

2

(−dpdz

)Re =

ρwDh

µ

Dh is the hydraulic diameter

Dh =4× cross section area

wetted perimeter=

4A

P

and w is the average velocity in the cross section

w =1

A

∫A

wdA

The wetted perimeter depends on whether the full duct or quarter duct is being used in the numericalmodel. (See Figure 8.) For the full duct, P = 2(Lx + Ly). For the quarter duct, P = Lx + Ly.

Combining the expressions for f and Re yields

fRe =2D2

h

wµ

(−dpdz

). (29)

The numerical solution for the fully developed flow problem uses these steps.

1. Specify the geometry of duct cross section, Lx and Ly.

2. Choose arbitrary values of µ and −dp/dz.

3. Solve the numerical model of the diffusion equation to obtain wi,j on the mesh.

4. Post-process:

a. Compute w from wi,j and given mesh parameters.

b. Compute fRe from Equation (29)

c. Create plots of the solution.

The CVFD solution for laminar flow in a rectangular duct is implemented in the demoFullyDevRectand demoFullyDevQuarterRect m-files.

7 Truncation Errors

The truncation error for the approximation leading to Equation (9) is O((∆x)2) +O((∆y)2). Con-sider the case where the solution is obtained on a square domain with a uniform grid. Then

∆x = ∆y = h

If φex is the exact solution and φfd is the finite- difference solution then at any point in the domain

e ≡ φex − φfd ∼ O(h2) (30)

Since e is defined wherever φex and φfd are defined it is also a (discrete) field variable. If e is somesuitably defined average nodal error then we also expect that

e ∼ O(h2) (31)

REFERENCES 16

nx = ny N‖e‖2N

reduction in‖e‖2N

from

the next coarser grid

5 25 1.190× 10−2 —

10 100 1.420× 10−31.190

0.1420= 8.39

20 400 1.755× 10−41.420

0.1755= 8.09

40 1600 2.187× 10−51.755

0.2187= 8.02

80 6400 2.714× 10−62.187

0.2732= 8.06

160 25600 3.414× 10−72.732

0.3414= 8.00

320 102400 4.267× 10−83.414

0.4267= 8.00

Table 6: Reduction in normalized error for the control-volume finite-difference solution to modelproblem 1.

With these estimates of truncation error we can use model problem 1, which has an easilycomputable exact solution, to check the correctness of computer codes that implement the control-volume finite-difference method. After obtaining the numerical solution we compute the normalizedL2 norm of the error at each node

‖e‖2N

=

√∑e2i

N(32)

Substitute the average error, e for ei

‖e‖2N∼√∑

e2

N=

√Ne2

N=

e√N

For a uniform grid on a square domain N = n2x, where nx is the number of control volumes in thex-direction (cf. Figure 1). The grid spacing is h = Lx/(nx + 1). Thus

‖e‖2N∼ O(h2)

nx=O((Lx/(nx + 1))

2)

nx∼ O

(1

nx

)3

(33)

If the code is working correctly, doubling the number of control volumes in each direction willreduce the normalized truncation error defined in Equation (32) by a factor of eight. The results oftesting a correct code are given in Table 6. The data in the first and last column of the table areconsistent with Equation (33).

References

[1] Jack J. Dongarra, Iain S. Duff, Danny C. Sorensen, and Henk A. van der Vorst. Solving LinearSystems on Vector and Shared Memory Computers. SIAM, Philadelphia, 1991.

REFERENCES 17

[2] Gene Golub and James M. Ortega. Scientific Computing: An Introduction with Parallel Com-puting. Academic Press, Inc., Boston, 1993.

[3] S.V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington D.C., 1980.

10−3

10−2

10−1

100

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

∆ x

Measure

d e

rror

Measured

Theoretical error ~ (∆ x)3

Figure 9: Reduction in measured truncation error with mesh size for model problem 1. Numericaldata for this plot is from Table 6.

A AXISYMMETRIC GEOMETRY 18

P

δxw

δxe

S

W E

N

∆x

∆r

xe

xw

rn

rs

δrs

δrn

P

S

N

W

E

Figure 10: Axisymmetric control-volume.

A Axisymmetric Geometry

The diffusion equation in axisymmetric coordinates is

∂

∂x

(Γ∂φ

∂x

)+

1

r

∂

∂r

(1

rΓ∂φ

∂r

)+ S = 0 (34)

Integrating the first term in Equation (34) over the finite control volume yields∫ rn

rs

∫ xe

xw

∂

∂x

(Γ∂φ

∂x

)dx rdr =

∫ rn

rs

[(Γ∂φ

∂x

)e

−(

Γ∂φ

∂x

)w

]dy

≈[(

Γ∂φ

∂x

)e

−(

Γ∂φ

∂x

)w

]r∆r

≈[ΓeφE − φPδxe

− ΓwφP − φWδxw

]∆r

Integrating the second term in Equation (34) gives∫ xe

xw

∫ rn

rs

1

r

∂

∂r

(rΓ∂φ

∂r

)r dr dx ≈

[rnΓn

φN − φPδrn

− rsΓsφP − φSδrs

]∆x

Integrating the source term in Equation (34) over the control volume gives∫ xe

xw

∫ rn

rs

S r dr dx ≈ SP ∆x rp∆r (35)

where rp is the radial position of node P , and Sp is the source term evaluated at node P .