ME - 733Computational Fluid Mechanics

Lecture 3

Dr./ Ahmed Nagib Elmekawy Oct 28, 2018

Classification of Partial Differential Equations

Refer to

Ch. 1Hoffmann, A., Chiang, S., Computational Fluid Dynamics for Engineers, Vol. I, 4th ed., Engineering Education System, 2000.

Ch. 2Pletcher, R. H., Tannehill, J. C., Anderson, D., Computational Fluid Mechanics and Heat Tranfer, 3rd ed., CRC Press, 2011.

• Physical problems are governed by many PDEs

• Some are governed by first order PDEs

• Numerous problems are governed by second order PDEs

• A few problems are governed by fourth-order PDEs.

Partial Differential Equations - Background

Examples

2

2

x

TC

t

T:)D1(EquationConductionHeat

=

−

+

=

−

2

2

2

2

y

T

x

TC

t

T:)D2(EquationConductionHeat

0yx

:EquationLaplaceD22

2

2

22 =

+

=−

)y,x(fyx

:EquationPoissonD22

2

2

22

=

+

=−

Examples (contd.)

2

2

2

2

x

uC

t

u:EquationWaveD1

=

−

+

=

−

2

2

2

2

2

2

y

u

x

uC

t

u

:)membranevibrating(EquationWaveD2

)plateVibrating(02

t

u2

h4

y

u4

2y

2x

u4

24

x

u4

D

)beamVibrating(04

x

u4

C2

t

u2

:EquationsthOrder4

=

+

+

+

=

+

There are 6 basic classifications:

(1) Order of PDE

(2) Number of independent variables

(3) Linearity

(4) Homogeneity

(5) Types of coefficients

(6) Canonical forms for 2nd order PDEs

Classification of Partial Differential Equations (PDEs)

The order of a PDE is the order of the highest partial derivative in the equation.

Examples:

(1st order)

(2nd order)

(3rd order)

(1) Order of PDEs

2

2

x

u

t

u

=

x

u

t

u

=

xsinx

uu

t

u3

3

+

=

Examples:

(2 variables: x and t)

(3 variables: r, q, and t)

(2) Number of Independent Variables

2

2

x

u

t

u

=

2

2

22

2u

r

1

r

u

r

1

r

u

t

u

q

+

+

=

(3) Linearity

PDEs can be linear or non-linear.

A PDE is linear if the dependent variable and all its derivatives appear in alinear fashion (i.e. they are not multiplied together or squared for example:

Examples:

(Non Linear)

(Non-linear)

(Linear)

(Non-linear)

(Non-linear)

(Linear)

(Non-linear)

tsinx

ue

t

u2

2t

2

2

+

=

−

0y

uy

x

u2

2

2

2

=

+

0t

u

x

uu

2

2

=

+

0uy

uy

x

ux

2 =+

+

y2

2

2

eusinx

u

x

u=+

+

xsiny

u

yx

u2

x

u2

22

2

2

=

+

+

1y

uu

x

u2

=

+

A PDE is called homogenous if after writing the terms in order,the right hand side is zero.

Examples:

(Non-homogeneous)

(Homogeneous)

(Homogeneous)

(4) Homogeneity

)y,x(fy

u

x

u2

2

2

2

=

+

0t

u

x

u2

2

=

−

ux

u

t

u2

2

2

2

=

−

(Non-homogeneous)

(Homogeneous)

5ut

u

x

u−=

+

5ut

)5u(

x

)5u(−=

−+

−

(5) Types of Coefficients

• If the coefficients in front of each term involving the dependent variable and its derivatives are independent of the variables (dependent or independent), then that PDE is one with constant coefficients.

(Variable coefficients)

(C constant; constant coefficients)

Examples

0y

ux

x

u2

22

2

2

=

+

0t

uC

x

u2

2

2

2

=

−

GFuy

uE

x

uD

y

uC

yx

uB

x

uA

2

22

2

2

=+

+

+

+

+

• (Standard Form)

• where A, B, C, D, E, F, and G are either real constants or real-valued functions of x and/or y.

• The equations of the characteristics in physical space are given by:

• Depending on the value of (B2-4AC), characteristic curves can be real or imaginary.

• For problems in which real characteristics exist, a disturbance can propagate only over a finite region.

(6) Canonical forms for 2nd order PDEs (Linear)

3. Classification of partial differential equations:

Lets consider the following partial differential equation:

2 2 2

2 2d e f g

x x y y xba

yc

+ + + + + =

2nd order derivative 1st order derivative

Appropriate values to the various coefficients a,b,c,d,e,f and

g result to various known equations, including the N-S:

e.g_1: a= 1

c=-1

b=d=e=f=g=0

2 2

2 2t x

=

Wave equation

e.g_2: e = 1

c=-1

a= b=d=f=g=0

2

2t x

=

Diffusion equation

e.g_3: a = 1

c= 1

e= b=d=f=g=0Laplace equation

2 2

2 20

x y

+ =

e.g_4: a= 1-M2

c= 1

b=d=e=f=g=0

2 22

2 2(1 ) 0M

x y

− + =

Compressible Isentropic flow ,

where represents the potential

of the flow

2 2 2

2 2d e f g

x x x y xba

yc

+ + + + + =

2nd order derivative 1st order derivative

The above equation can only be one of the following types:

➢Elliptic type, if : b2-ac<0

➢Parabolic type, if : b2-ac=0

➢Hyperbolic type, if : b2-ac>0

2 2

2 2t x

=

t

(x)

x

Wave equation:

Hyberbolic type

A BC

➢No ‘information’ exists at point B before the arrival

of point A to B, and no information at point C before

the arrival of point B to C

➢The information propagates with a characteristics

velocity, which is the speed of the wave

t+dt

t+2dt

➢The information at B depends only on the information at D!!!

➢The DB line is called the ‘characteristic’ line of the flow field

2 22

2 2(1 ) 0M

x y

− + =

➢Hyperbolic type: Flow around planes at M>=1

➢Parabolic type:

1-D Boundary layer, heat convection, diffusion processes,

etc

2

2t x

=

➢The information at B depends only on the information at A!!!

➢Elliptic type:

Boundary dependant processes

➢The information at B depends both on A (upstream) and G

(downstream) of point B!,

➢i.e. it depends on the values of the flow field at the boundary

(e.g. upstream and downstream pressures Pup and Pdown determine

flow rate, etc)

Pup Pdown

➢In most cases, the differential equations expressing a

physical process, can not be solved with simple numerical

techniques described.

➢The usual numerical procedure then is to express this

differential equations in finite differences.

➢This implies that the various derivatives are expressed as a

function of values at discrete points located around the point

at which the derivative is to be estimated

➢This can be usually achieved by using the Taylor’s rule for

each derivative appearing in the equation

S

E

N

PW

Discretisation of the geometry:

➢The physical domain on which we are looking for the

solution of the differential equation, has to be ‘discretised’

into a number of ‘nodes’ or ‘grid’ or ‘mesh’ points, which

have to describe the boundaries of this domain

Point of the numerical solution or grid point

West East

North

South

Individual terms (partial derivatives):

Any function can be approximated using truncated Taylor

series expansions:

( ) ( )f z f zf

z

z f

z

z f

z

z f

z

z z

z z

= + +

+ +

= =

= =

02

3 4

0

2 2

2

0

3 3

3

0

4 4

4

0

!

! !... +

xPW Ei-1 i+1i

Dx Dx

Considering only 1-direction:

We can relate conditions at i+1 (E) and i-1 (W) to those at i (P)

using Taylor series expansions; i.e.,

( )

( ) ( )

i i

i i

i i

xx

x

x

x

x

x

x

+ = + +

+ +

1

2 2

2

3 3

3

4 4

4

2

6 24

DD

D D + ...

This Equation can be rearranged in terms of the first derivative:

( ) ( )

x x

x

x

x

x

x

x

i

i i

i

i i

=−

−

− − −

+1

2

2

2 3

3

3 4

4

2

6 24

D

D

D D ...

➢An algebraic model of the first derivative can be generated

from the above by dropping all but the first term on the right-

hand side (truncating); i.e.:

x xi

i i−+1

D➢The above is a two-point forward difference approximation

to the first derivative:

xPW Ei-1 i+1i

Dx Dx

( )

( ) ( )

i i

i i

i i

xx

x

x

x

x

x

x

− = − +

− + +

1

2 2

2

3 3

3

4 4

4

2

6 24

DD

D D ...

➢It is said to be first-order accurate, or that it has a truncation

error of order Dx [O(Dx)]. That is, the leading term of all those

dropped from the full Taylor series included Dx to the first

power (Note Dx< 1).

➢The function can be also expressed by considering not the

‘East’ point but the ‘West’ point:

x xi

i i− −1

D

➢Which can result to another two-point backward difference

approximation to the first derivative:

xPW Ei-1 i+1i

Dx Dx

➢A more accurate approximation to the first derivative can

be obtained by combining the forward and the backward

discretisation schemes:

( )

x x

x

xi

i i

i

=−

− ++ −1 1

2 3

32 6D

D...

or, truncating:

x xi

i i−+ −1 1

2D

➢The above is the three-point central difference

approximation to the first derivative

• The downstream region affected by a disturbance at point A is called the zone of influence (indicated by horizontal shading).

• A signal at point A will be felt only if it originated from a finite region called the zone of dependence of point A (vertical shading).

Figure 1-1. Zone of influence (horizontal shading) and zone of dependence (vertical shading) of point A.

• The equation is then classified as follows:

Classification of PDE

PDE is Elliptic− 0AC4B2

PDE is Hyperbolic− 0AC4B2

PDE is Parabolic=− 0AC4B2

• Parabolic PDE solution “propagates” or diffuses

• Hyperbolic PDE solution propagates as a wave

• Elliptic PDE equilibrium

This terminology of elliptic, parabolic, and hyperbolic, reflect the analogybetween the standard form for the linear, 2nd order PDE and conic sectionsencountered in analytical geometry:

for which when one obtains the equation for an ellipse,

when one obtains the equation for a parabola,

and when one gets the equation for a hyperbola.

0FEyDxCyBxyAx22 =+++++

0AC4B2 −

0AC4B2 =−

04ACB2 −

• A partial differential equation is elliptic in a region if (B2 - 4AC) < 0 at all points of the region.

• An elliptic PDE has no real characteristic curves.

• A disturbance is propagated instantly in all directions within the region.

• Examples of elliptic equations are Laplace's equation

• (1-9)

• and Poisson's equation

• (1-10)

• The domain of solution for an elliptic PDE is a closed region, R.

• On the closed boundary of R, either the value of the dependent variable, its normal gradient, or a linear combination of the two is prescribed.

• Providing the boundary conditions uniquely yields the solution within the domain.

Elliptic Equations

Figure 1-2. The domain of solution for an elliptic PDE.

Figure 1-3. The domain of solution for a parabolic PDE.

• A PDE is classified as parabolic if (B2 - 4AC) = 0 at all points of the region.

• The solution domain for a parabolic PDE is an open region.

• For a parabolic PDE there exists one characteristic line.

• 1-D Unsteady heat conduction

• and diffusion of viscosity, expressed as

are examples of parabolic PDEs.

• An initial distribution of the dependent variable and two sets of boundary conditions are required for a complete description of the problem.

• The BCs are prescribed as the value of the dependent variable or its normal derivative or a linear combination of the two.

• The solution of the parabolic equation marches downstream within the domain from the initial plane of data satisfying the specified BCs.

Parabolic Equations

• A PDE is called hyperbolic if (B2 - 4AC) > 0 at all points of the region.

• A hyperbolic PDE has two real characteristics.

• An example of a hyperbolic equation is the second-order wave equation:

• A complete description of the flow governed by a second-order hyperbolic PDE requires two sets of initial conditions and two sets of boundary conditions.

• The initial conditions are those at t = 0.

Hyperbolic Equations

(a)

Here, A=1, B=0, C=2, D=E=F=G=0 B2-4AC = 0 - 4(1)(2) = -8 < 0 thisequation is elliptic.

(b)

Here, A=1, B=0, C=-2, D=E=F=G=0 B2-4AC = 0 - 4(1)(-2) = 8 > 0 thisequation is hyperbolic.

(c)

Here, A=1, B=0, E=-2, C=D=F=G=0 B2-4AC = 0 - 4(1)(0) = 0 this equation is parabolic.

Examples

0y

u2

x

u2

2

2

2

=

+

0y

u2

x

u2

2

2

2

=

−

0y

u2

x

u2

2

=

−

(d)

Here, A=1, B=-4, C=1, D=E=F=G=0 B2-4AC = 16 - 4(1)(1) = 12 > 0 thisequation is hyperbolic.

(e)

Here, A=3, B=-4, C=-5, D=E=F=G=0 B2-4AC = 16 - 4(3)(-5) = 76 > 0 thisequation is hyperbolic.

(f)

Here, A=3, B=-4, C=-5, D=8, E=-9, F=6, G=27exy B2-4AC = 16 - 4(3)(-5) > 0 this equation is hyperbolic.

Examples

0y

u

yx

u4

x

u2

22

2

2

=

+

−

xy

2

22

2

2

e27u6y

u9

x

u8

y

u5

yx

u4

x

u3 =+

−

+

−

−

0y

u5

yx

u4

x

u3

2

22

2

2

=

−

−

(g)

Here, A=y, B=0, C=-1, D=E=F=G=0 B2-4AC = 0 - 4(y)(-1) = 4y for y>0, thisequation is hyperbolic;

for y=0, this equation is parabolic;

for y<0, this equation is elliptic.

Examples

0y

u

x

uy

2

2

2

2

=

−

Hyperbolic

x

y

EllipticParabolic

(h)

Here, A=1, B=2x, C=1-y2, D=E=F=G=0 B2-4AC = 4x2 - 4(1)(1-y2) = 4x2+4y2-4or x2+y2 >,=,< 0

Examples

0y

u)y1(

yx

ux2

x

u2

22

2

2

2

=

−+

+

Elliptic

Hyperbolic

x

y

Parabolic on

surface of circle

(i)

Here, A=1, B=-y, C=0, D=E=F=G=0 B2-4AC = y2

for y=0, this equation is parabolic;

for y0, this equation is hyperbolic.

Examples

0uy

uy

x

ux

yx

uy

x

u2

2

2

=+

+

+

−

Hyperbolic

Hyperbolic

x

y

Parabolic

(j)

Here, A=sin2x, B=sin2x, C=cos2x, D=E=F=G=0 B2-4AC = sin22x-4sin2xcos2x =4sin2xcos2x-4sin2xcos2x = 0 this equation is parabolic everywhere.

Example

xy

u)x(cos

yx

u)x2(sin

x

u)x(sin

2

22

2

2

22 =

+

+

(k)

This must be converted to 2nd order form first:

and

subtracting,

Now, A=1, B=0, C=-1, D=E=F=G=0 B2-4AC=4 > 0

Hyperbolic.

0x

u

t

u=

+

0xt

u

t

u2

2

2

=

+

0x

u

tx

u2

22

=

+

0x

u

t

u2

2

2

2

=

−

(l)

Again, convert to 2nd order form first:

and

adding,

Again, A=1, B=0, C=-1, D=E=F=G=0 B2-4AC = 4 > 0 Hyperbolic.

Example

0x

u

t

u=

−

0xt

u

t

u2

2

2

=

−

0

x

u

tx

u2

22

=

−

0x

u

t

u2

2

2

2

=

−

Classify the steady 2-D velocity potential equation.

• Solution: A= (1-M2), B = 0, and C = 1 Thus, (B2-4AC) = -4(1-M2).

• If M < 1 (subsonic flow), then (B2-4AC) < 0 and the equation is elliptic.

• For M = 1 (sonic flow), (B2 - 4AC) = 0 and the equation is parabolic.

• For M > 1 (supersonic flow), (B2 - 4AC) > 0 and the equation is hyperbolic.

• Assume that a body moving with a velocity u in an inviscid fluid is creating disturbances which propagate with the speed of sound, a.

• If the velocity u is smaller than a, that is, if the flow is subsonic, then the disturbance is felt everywhere in the flow field.

• As the speed of the body u increases and approaches the speed of sound, a front is developed, with a region ahead of it which does not feel the presence of the disturbance.

• This region is known as the zone of silence. Thus the disturbance is felt only behind the front. This region is known as the zone of action.

Example 1.1

Figure 1-4a. Propagation of disturbance in subsonic flow.

Figure 1-4b. Propagation of disturbance in sonic flow.

• When the speed u is further increased, to the extent that it exceeds the speed of sound, a conical front (in three-dimensional analysis) is formed.

• The effect of the disturbance is felt only within this cone.

• This conical front is known as the Mach cone in three-dimensional space or as Mach lines in two-dimensional space.

• Mach lines patch two different solutions of the PDE and thus represent the characteristics of the PDE.

Figure 1-4c. Propagation of disturbance in supersonic flow.This

Model equations• Several partial differential equations will be used as model equations.

• These equations will be used to illustrate the application of various finite differencing techniques and stability analyses.

• By observing and analyzing the behavior of the numerical methods when applied to simple model equations, an understanding should be developed which will be useful in studying more complex problems.

• The selected equations are primarily derived from principles of fluid mechanics and heat transfer.

• The selected model PDEs are as follows:

4. The y-component of the Navier-Stokes equation reduced to Stokes' first problem:

System of First-Order PDEs• The equations of fluid motion are composed of conservation of mass,

conservation of momentum, and conservation of energy.

• The governing equations may be expressed by partial differential equations, thus forming a system of second-order PDEs.

• For certain classes of problems, the governing equations are reduced to a system of first-order PDEs.

• For example, the equations of fluid motion for inviscid flow fields, known as the Euler equations, belong to this category.

• Furthermore, in some applications a higher-order PDE may be reduced to a system of first-order PDEs by introducing new viariables.

• In this section, the conditions under which a system of first-order PDEs is classified will be explored.

System of First-Order PDEs• Consider a set of first-order PDEs expressed in the following form:

• where represents a vector (or column matrix) containing the unknown variables.

• The elements of the coefficient matrices [A] and [B] are functions of x, y, and t; and the vector is a function of , x, and y.

• For example, a set of two first-order PDEs could be represented by the following equations:

• If the eigenvalues of the matrix [A] are all real and distinct, the set of equations is classified as hyperbolic in t and x.

• For complex eigenvalues of [A], the system of equations is elliptic in t and x.

• Similarly, the set of equations is hyperbolic in t and y if all the eigenvalues of matrix [B] are real and distinct; otherwise, for complex eigenvalues, the set of equations is classified as elliptic.

• If the system of equations has the following form

Where

then the set of equations is classified according to the sign of

Where

Example 1.2 Classify the following system of partial differential equations.

• Now, H is determined as H = R2- 4PQ = -4.

• Since H is negative, the system is classified as elliptic.

How do you calculate eigenvalues?

det(A- I )=|A- I |=0 for a 2x2 matrix:

−

−=

−

=−

10

01

10

01

10

01IA

( )( )

( ) 110

01110

01det

2=−=

=−−=−

−=−

IA

The eigenvalues of the matrix [A] are all real but not distinct, the set of equations is classified as parabolic in t and x.

How do you calculate eigenvalues?

det(B- I )=|B- I |=0 for a 2x2 matrix:

−−

−=

−

−=−

1

1

10

01

01

10IB

( )

complexi

B

=−=−=

=+=−−

−=−

11

011

1det

2

2

I

Similarly, the set of equations is hyperbolic in t and y if all the eigenvalues of matrix [B] are real and distinct; otherwise, for complex eigenvalues, the set of equations is classified as elliptic.

How do you calculate eigenvectors and eigenvalues?

det(A- I )=|A- I |=0 for a 2x2 matrix:

−

−=

−

=−

2221

1211

2221

1211

10

01

aa

aa

aa

aaIA

( )( )

( ) 2

221121122211

21122211

2221

1211

0

0det

++−−=

=−−−=−

−=−

aaaaaa

aaaaaa

aaIA

For a 2-dimensional problem such as this, the equation above is a simple quadratic equation with two solutions for . In fact, there is generally one eigenvalue for each dimension, but some may be zero, and some complex.

Eigen values and Eigen Vectors

The solution to E.05 is:

0 = a11a22 − a12a21 − a11 + a22( ) + 2

= a11 + a22( )a11 +a22( )

2

4 a11a22 − a12a21( )

(E.06)

This “characteristic equation” does not involve x, and the resulting values of can be used to solve for x.

Consider the following example:

A =1 2

2 4

(E.07)

Eqn. E.07 doesn’t work here because a11a22-a12a12=0, so we use E.06:

0 = a11a22 − a12a21 − a11 + a22( ) + 2

0 =1 4 − 2 2 − (1+ 4) + 2

(1+ 4) = 2

We see that one solution to this equation is =0, and dividing both sides of the above equation by yields =5.

Thus we have our two eigenvalues, and the eigenvectors for the first eigenvalue, =0 are:

Ax = x, A− I( )x = 0

1 2

2 4

−

0

0

x

y

=

1 2

2 4

x

y

=

1x+ 2y

2x+ 4y

=

0

0

These equations are multiples of x=-2y, so the smallest whole number values that fit are x=2, y=-1

For the other eigenvalue, =5:

1 2

2 4

−

5 0

0 5

x

y

=

−4 2

2 −1

x

y

=

−4x+ 2y

2x −1y

=

0

0

-4x + 2y = 0, and 2x − y = 0, so, x =1, y = 2

This example is rather special; A-1 does not exist, the two rows of A- I

are dependent and thus one of the eigenvalues is zero. (Zero is a

legitimate eigenvalue!)

The procedure is:

1) Compute the determinant of A- I

2) Find the roots of the polynomial given by | A- I|=0

3) Solve the system of equations (A- I)x=0

Alternative method• Recall that characteristics represent a family of lines across which the

properties are continuous.

• Now, define S to represent characteristic surfaces and normal to these surfaces denoted by 𝑛.

• In the Cartesian system, we may write (for 2-D problems)

•

• At this point, we seek a relation whereby the number of possible characteristics may be determined.

• If the characteristic normals are all real, then the system is classified as hyperbolic. If they are complex, then the system is elliptic. For mixed real and complex values, the system is mixed elliptic/hyperbolic.

• The system is classified as parabolic if there is less than K real characteristics, where K is the number of PDEs in the system.

• To introduce the required relation, consider the following model equation:

• A wave-like solution (characteristics direction) for the system may be obtained if ITI = 0 where

• Now, matrix [T] is formed as

• from which the determinant is computed as

• Divide by n2x to obtain

• This equation may be written as

• Note that the notation previously defined for Equation (1-23) is used in the equation above.

• Therefore it is seen that, if H > 0, the system is hyperbolic; if H < 0, the system is elliptic; and if H = 0, the system is parabolic.

For this purpose, [T] isdetermined as

Example 1.2 Classify the following system of partial differential equations.

• from which both values of (ny/nx) are imaginary and, therefore, the system is elliptic.

Back to example 1-2For this purpose, [T] is determined as

Example 1.3• The governing nondimensional equations of fluid motion for steady, inviscid

and incompressible flow in two dimensions are given by: Classify the system of equations.

• Therefore, the vector formulation is written as

Select the unknown vector, Q, as

Solution

• Since mixed real and complex values result, the system is a mixed hyperbolic/elliptic system.

• The governing equations of motion for one-dimensional, inviscid flows are given by the Euler equations.

• If the assumption of perfect gas is imposed, the system is written as:

• It is required that this system be classified.

• Solution: Define the variable vector Q as:

• The system is written in the vector form as

Example 1.4

• The eigenvalues of the system are obtained from

where

• from which

• Since all the eigenvalues are real, the system is classified as hyperbolic.

solution

• In order to obtain a unique solution of a PDE, a set of supplementary conditions must be provided to determine the arbitrary functions which result from the integration of the PDE.

• Those conditions are classified as boundary or initial conditions.

• An initial condition is a requirement for which the dependent variable is specified at some initial state.

• A boundary condition is a requirement that the dependent variable or its derivative must satisfy on the boundary of the domain of the PDE.

• Various types of boundary conditions which will be encountered are:

1. The Dirichlet boundary condition. If the dependent variable along the boundary is prescribed, it is known as the Dirichlet type.

2. The Neumann boundary condition. If the normal gradient of the dependent variable along the boundary is specified, it is called the Neumann type.

3. The Robin boundary condition. If the imposed boundary condition is a linear combination of the Dirichlet and Neumann types, it is known as the Robin type.

4. The Mixed boundary condition. Frequently the boundary condition along a certain portion of the boundary is the Dirichlet type and, on another portion of the boundary, a Neumann type. This type is known as a mixed boundary condition.

Initial and Boundary Conditions

• As an example, consider transient conduction in two-space dimensions.

• Assume that a long rectangular bar has been heated to a temperature distribution of T =f(x, y).

• An initial condition would then be prescribed such that for t = 0, T = f(x,y)

• Now, place the bar in an environment in which:

• the lower and right sides are in contact with a convecting fluid of temperature Tf and a constant film coefficient of h,

• while the left side is insulated (adiabatic) and

• the upper side is kept at a constant temperature.

• The corresponding boundary conditions are:

• These are shown in Figure 1-5.

Figure 1-5. Sketch illustrating the imposed boundary conditions on the rectangular bar.

and

• In order to solve a given PDE by numerical methods, the partial derivatives in the equation are approximated by finite difference relations.

• The resulting approximate equation, which represents the original PDE, is called a finite difference equation (FDE).

• Consider a 2-D rectangular domain. We wish to solve a PDE within this domain subject to imposed initial and boundary conditions.

• The rectangular domain is divided into equal increments in the x and y directions.

• Denote the increments in the x direction by Δx and the increments in they direction by Δy. Note that the increments in the x direction do not need to be equal to the increments in the y direction.

• These increments may be defined as mesh size, step size, or grid size.

• The location of mesh points, grid points, or nodes is designated by i in the x direction and by j in the y direction.

• The maximum number of grid points in the x and y directions are denoted by I M and J M, respectively.

• The finite difference equation that approximates the PDE is an algebraic equation.

• This algebraic equation is written for each grid point within the domain.

Remarks and Definitions

• The solution of the finite difference equations provides the values of the dependent variable at each grid point.

• The objectives are to study the various schemes to approximate the PDEs by finite difference equations and to explore numerical techniques for solving the resulting approximate equations.

Remarks and Definitions

Figure 1-6. Sketch illustrating the nomenclature of computational space.

• Before proceeding with an analysis of numerical techniques, it is necessary to define additional terminology for concepts which will be investigated in the upcoming chapters.

1. Consistency: A finite difference approximation of a PDE is consistent if the finite difference equation approaches the PDE as the grid size approaches zero.

2. Stability: A numerical scheme is said to be stable if any error introduced in the finite difference equation does not grow with the solution of the finite difference equation.

3. Convergence: A finite difference scheme is convergent if the solution of the finite difference equation approaches that ofthe PDE as the grid size approaches zero.

4. Lax's equivalence theorem: For a FDE which approximates a well-posed, linear initial value problem, the necessary and sufficient condition for convergence is that the FDE must be stable and consistent.

Remarks and Definitions