Chapter C Finite Difference Method

8/10/2019 Chapter C Finite Difference Method

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Professor Christopher Chukwutoo Ihueze(B.Eng,M.Eng,Ph.D.)

Department of industrial and production engineering,

Nnamdi Azikiwe University,Awka

Faculty of Engineering,Nnamdi Azikiwe University,Awka


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Lecture Note: this lecture note is available for download in pdfformat at www.ccihueze.com.

Recommended Textbooks:

Grading: Attendance (10%)

Bi-Weekly homework (12%)

Midterm (28%)

Final Exam (50%)

This chapter involves formation of difference equation, Meshing

and difference equation formulation,
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Introduction

Finite Difference Modelling means discretizing a field

function and deriving a difference equation of thefunction or model to be approximated

A differenceequation model of a

field function is

passed through

interior mesh pointsof a region to

establish a system of

equations


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Meshing and difference equation formulation

h 2h

10h

h

8h

Figure 1: Finite differencemodel of a field region

Supposing

represents a regular partition of interval [a,b]

Where I = 0. 1 2, ., n and

The points

are called interior mesh points of the interval [a,b]

By expressing differential equation as

and by letting


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Meshing and difference equation formulation CONTINUES

So that equation (4) becomes

Or by rearrangement

Equation (10) gives the finite difference equation which is an

approximation to the differential equation.

and by replacing by their central difference approximations derived as


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Example B.1 CONTINUES

Eq(3) is re-written in the form of eq(1),

It is recognized that

Substituting into the difference equation (2) and simplifying gives

Interior points are as given as,


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Which on substitution into equation (5) for i= 1, 2, 3, 4, 5, 6, 7 results in

the system of equations


This can be put in matrix form,

Where A is the matrix of coefficients, Y is the solution vector and B is acolumn matrix of constants. If each side of equation 6 is multiplied by

then

The matrices are as displayed below


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Thus

This becomes


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The matrix multiplication results in the solution vector Ybeing equal to

The solution by finite difference method to the boundary value problem

is presented in tabular form Table 1:

i xi yi

1 1.125 -0.19852 1.250 -0.4165

3 1.375 -0.6507

4 1.500 -0.8988

5 1.625 -1.1591

6 1.750 -1.42987 1.875 -1.7103


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B: SOLUTION OF LAPACE EQUATION

Example B. 2:Approximate the solution of a Laplace equation subject to the following

boundary conditions;

Use the difference equation of Laplace equation as

SOLUTION: The problem is thus graphically presented

11u

21u

12u

22u

x

y

0

0 0

0

0

0

8

16The finite difference equation applicable to

this boundary value problem (Laplace

equation) is

Application of equation (1) results in a system of linear equations put in

matrix form as


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By employing Gauss-Seidel scheme

B: SOLUTION OF LAPACE EQUATION Continues

U0 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12

6 6 4.75 4.25 3.938 3.813 3.734 3.703 3.684 3.676 3.671 3.669 3.668

4 3.5 2.5 1.875 1.625 1.469 1.406 1.367 1.352 1.342 1.338 1.335 1.334

12 7.5 6.5 5.875 5.625 5.469 5.406 5.367 5.352 5.342 5.338 5.335 5.334

8 4 2.75 2.25 1.938 1.813 1.734 1.703 1.684 1.676 1.671 1.669 1.668

The solution thus becomes


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Example B.3:Solve the Laplace equation subject to the boundary conditions

SOLUTION: For better picture the interior points and boundary conditionsare depicted thus

y

x

25

50

75

50 7525

13u 23u

33u

12u 22u 32u

11u

21u 31u

0

0

0 0 0

0

A mesh size h(=1/4)utilized in equation(B.1), the

system of equations that results put in matrix form

is

l B 3 i


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Example B.3: continues

By employing Gauss-Seidel schemeU0 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 U16 U17 U18 U19

6 6.25 6.38 8.61 6.41 7.46 6.33 6.86 6.30 6.56 6.28 6.4 6.384 6.325 6.317 6.288 6.283 6.269 6.267 6.259

13 12.75 17.25 12.81 14.92 12.18 13.12 12.59 13.11 12.55 12.80 12 .57 8 1 2. 65 1 2. 61 12 .57 5 1 2. 56 4 1 2. 538 12 .5 33 1 2. 51 9 1 2. 517

20 19.5 19.13 21.20 18.94 19.97 18.85 19.36 18.80 19.05 18.88 1 8. 9 1 8. 789 18 .8 25 18 .80 5 1 8. 78 8 1 8. 782 18 .7 69 1 8. 76 7 1 8. 759

12 12.75 17.19 12.81 14.92 12.68 13.72 12.59 13.11 12.55 12.80 12 .95 8 1 2. 65 12 .6 58 12 .57 5 12 .57 1 2. 538 12 .5 34 1 2. 51 9 1 2. 517

25 43.25 25.75 29.88 25.38 27.44 25.19 26.22 25.1 25.59 25.15 2 5. 3 2 5. 268 2 5. 15 25 .13 4 2 5. 07 5 2 5. 067 25 .0 38 2 5. 03 3 2 5. 019

40 38.75 42.57 37.94 39.96 37.70 38.72 37.60 38.07 37.55 37.80 37 .57 8 3 7. 65 3 7. 61 37 .57 5 3 7. 56 4 3 7. 538 37 .5 33 3 7. 51 9 3 7. 517

20 19.25 19.13 21.17 18.94 19.97 18.85 19.36 18.80 19.05 20.40 1 8. 9 1 8. 979 18 .8 25 18 .82 9 1 8. 78 8 1 8. 785 18 .7 69 1 8. 76 7 1 8. 759

40 38.75 42.5 37.94 39.96 37.70 38.72 37.60 38.07 37.55 37.80 37 .95 8 3 7. 65 37 .6 58 37 .57 5 37 .57 3 7. 538 37 .5 34 3 7. 51 9 3 7. 517

60 57.5 56.88 58.77 56.47 57.48 56.35 56.86 56.30 56.54 56.28 5 6. 4 5 6. 384 56 .3 25 56 .31 7 5 6. 28 8 5 6. 283 56 .2 69 5 6. 26 7 5 6. 259

The solution vector is thus displayed

B SOLUTION OF LAPACE EQUATION C i


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Example B.4:

Solve Example B.3 but with a modified boundary condition as


The boundary conditions together with the interior mesh points are as

depicted in the diagram below

11p

21p

31p

33p

13p

12p 22p 32p

23p

y

x

20

20

40

50

60

70

20

20 4010

10 30

With a mesh size of 0.25, the use of equation B.1

results in the system of equations;

By employing Gauss-Seidel schemeU0 U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14 U15 U16 U17 U18

27.5 27.5 25.627 24.844 23.907 23.398 22.93 22.661 22.427 22.291 22.173 22.105 22.047 22.012 21.983 21.966 21.951 21.936 21.931

35 35 33.75 32.188 31.211 30.313 29.78 29.316 29.045 28.811 28.675 28.557 28.489 28.431 28.396 28.367 28.35 28.327 28.319

42.5 42.5 41.875 40.938 40.156 39.629 39.18 38.909 38.677 38.54 38.423 38.355 38.297 38.262 38.233 38.216 38.201 38.186 38.181

45 37.5 35.625 33.438 32.383 31.406 30.864 30.391 30.118 29.883 29.746 29.629 29.561 29.502 29.468 29.438 29.421 29.398 29.391

50 45 41.25 39.063 37.188 36.094 35.156 34.609 34.141 33.867 33.633 33.496 33.379 33.311 33.252 33.218 33.188 33.157 33.148

55 52.5 50 48.438 47.305 46.406 45.854 45.391 45.117 44.883 44.746 44.629 44.561 44.502 44.468 44.438 44.421 44.398 44.391

32.5 30 26.875 25.625 24.531 23.965 23.477 23.201 22.964 22.827 22.709 22.641 22.582 22.548 22.519 22.502 22.487 22.471 22.467

45 40 36.875 34.688 33.477 32.5 31.938 31.465 31.19 30.955 30.818 30.7 30.632 30.573 30.539 30.51 30.493 30.47 30.462

57.5 52.5 50.625 49.219 48.281 47.695 47.227 46.948 46.714 46.577 46.459 46.391 46.332 46.298 46.269 46.252 46.237 46.221 46.217

The solution matrix becomes

B SOLUTION OF LAPACE EQUATION C ti


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Example B.5:Derive the finite difference equation replacement for the Poisson equation

SOLUTION: For square discretization of a plane

then the poisson equation becomes

Equation ( b1) is re-written in subscript form to become


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B SOLUTION OF LAPACE EQUATION C ti


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Approximate the solution of the Poisson equation


subject to the boundary condition

Solution: The boundary condition is displayed graphically as thusy

x

0

0

0

0 00

13u 23u

12u 22u 32u

11u

21u 31u

0

0

0 0 0

0 0

Making use of the derived difference equation(B.2), with fij = -64 the applicable finite

difference equation becomes

Equation 8.1 evaluated at the interior mesh points results in the system

of linear equations

Example B.6:

E l B 6 ti


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To preclude monotony of method the above system is solved by the use of

Cramers rule as presented below;

Example B.6 continues

To conserve space a compact notation could be use for the Cramers rule to

obtain the remaining solutions. For example

Where


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Example B.6 continues

The solution vector becomes


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GOO LU K

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Chapter C Finite Difference Method