1. Computational Modeling for Engineering MECN 6040
Professor: Dr. Omar E. Meza Castillo
Department of Mechanical Engineering

2. Best known numerical method of approximation
Finite differences
Best known numerical method of approximation

3. FINITE DIFFERENCE FORMULATION OF DIFFERENTIAL EQUATIONS
finite difference form of the first derivative
Taylor series expansion of the function f about the point x,
The smaller the x, the smaller the error, and thus the more accurate the approximation.

4. The big question: How good are the FD approximations?
This leads us to Taylor series....

5. EXPASION OF TAYLOR SERIES
Numerical Methods express functions in an approximate fashion: The Taylor Series.
What is a Taylor Series?
Some examples of Taylor series which you must have seen

6. General Taylor Series
The general form of the Taylor series is given by
provided that all derivatives of f(x) are continuous and exist in the interval [x,x+h], where h=∆x
What does this mean in plain English?
As Archimedes would have said, “Give me the value of the function at a single point, and the value of all (first, second, and so on) its derivatives at that single point, and I can give you the value of the function at any other point”

7. Example: Find the value of f(6) given that f(4)=125, f’(4)=74, f’’(4)=30, f’’’(4)=6 and all other higher order derivatives of f(x) at x=4 are zero.
Solution: x=4, x+h=6  h=6-x=2
Since the higher order derivatives are zero,

8. The Taylor Series (xi+1-xi)= h step size (define first)
Reminder term, Rn, accounts for all terms from (n+1) to infinity.

9. Zero-order approximation
First-order approximation
Second-order approximation

10. Example: Taylor Series Approximation of a polynomial Use zero- through fourth-order Taylor Series approximation to approximate the function:
From xi=0 with h=1. That is, predict the function’s value at xi+1=1
f(0)=1.2
f(1)=0.2 - True value

11. Zero-order approximation
First-order approximation

12. Second-order approximation

13. Third-order approximation

14. Fourth-order approximation

16. Taylor Series to Estimate Truncation Errors
If we truncate the series after the first derivative term
Truncation Error
First-order approximation

17. Numerical Differentiation
Forward Difference Approximation

18. Numerical Differentiation
The Taylor series expansion of f(x) about xi is
From this:
This formula is called the first forward divided difference formula and the error is of order O(h). 18

19. Or equivalently, the Taylor series expansion of f(x) about xi can be written as
From this:
This formula is called the first backward divided difference formula and the error is of order O(h). 19

20. A third way to approximate the first derivative is to subtract the backward from the forward Taylor series expansions:
This yields to
This formula is called the centered divided difference formula and the error is of order O(h2). 20

21. Numerical Differentiation
Forward Difference Approximation

22. Backward Difference Approximation

23. Centered Difference Approximation

24. Example: To find the forward, backward and centered difference approximation for f(x) at x=0.5 using step size of h=0.5, repeat using h=0.25. The true value is
h=0.5
xi-1=0 - f(xi-1)=1.2
xi=0.5 - f(xi)=0.925
Xi+1=1 - f(xi+1)=0.2

25. Forward Difference Approximation
Backward Difference Approximation

26. Centered Difference Approximation

27. h=0.25
xi-1=0.25 - f(xi-1)=
xi=0.5 - f(xi)=0.925
Xi+1=0.75 - f(xi+1)=
Forward Difference Approximation

28. Backward Difference Approximation
Centered Difference Approximation

29. FINITE DIFFERENCE APPROXIMATION OF HIGHER DERIVATIVE
The forward Taylor series expansion for f(xi+2) in terms of f(xi) is
Combine equations: 29

30. Solve for f ''(xi):
This formula is called the second forward finite divided difference and the error of order O(h).
The second backward finite divided difference which has an error of order O(h) is 30

31. The second centered finite divided difference which has an error of order O(h2) is31

32. High-Accuracy Differentiation Formulas
High accurate estimates can be obtained by retaining more terms of the Taylor series.
High-Accuracy Differentiation Formulas
The forward Taylor series expansion is:
From this, we can write 32

33. Substitute the second derivative approximation into the formula to yield:
By collecting terms:
Inclusion of the 2nd derivative term has improved the accuracy to O(h2).
This is the forward divided difference formula for the first derivative. 33

34. Forward Formulas 34

35. Backward Formulas 35

36. Centered Formulas 36

37. Example
Estimate f '(1) for f(x) = ex + x using the centered formula of O(h4) with h = 0.25.
Solution
From Tables 37

38. In substituting the values:38

39. Error
Truncation Error: introduced in the solution by the approximation of the derivative
Local Error: from each term of the equation
Global Error: from the accumulation of local error
Roundoff Error: introduced in the computation by the finite number of digits used by the computer 39

40. Introduction to Finite Difference
Numerical solutions can give answers at only discrete points in the domain, called grid points.
If the PDEs are totally replaced by a system of algebraic equations which can be solved for the values of the flow-field variables at the discrete points only, in this sense, the original PDEs have been discretized. Moreover, this method of discretization is called the method of finite differences.
(i,j)

41. y y u y x x x Discretization: PDE FDE Explicit Methods
Simple
No stable
Implicit Methods
More complex
Stables ∆x y
n+1 ∆y y n u
m,n y
n-1 x x x
m-1 m
m+1

45. Summary of nodal finite-difference relations for various configurations:
Case 1: Interior Node

46. Case 2: Node at an Internal Corner with Convection

47. Case 3: Node at Plane Surface with Convection

48. Case 4: Node at an External Corner with Convection

49. Case 5: Node at Plane Surface with Uniform Heat Flux

50. SOLVING THE Finite difference EQUATIONS
Heat Transfer Solved Problems

51. The Matrix Inversion Method

54. jacobi ITERATION method

60. GAUSS-SEIDEL ITERATION

63. Pre-specified % tolerance based on the knowledge of your solution
Error Definitions
Use absolute value.
Computations are repeated until stopping criterion is satisfied.
If the following Scarborough criterion is met
Pre-specified % tolerance based on the knowledge of your solution

64. Matrix Inversion Method
USIG EXCEL
Matrix Inversion Method
=MMULT(A7:C9,E2:E4)
=MINVERSE(A2:C4)

65. Jacobi Iteration Method using Excel

66. Gauss-Seidel Iteration Method using Excel

67. A large industrial furnace is supported on a long column of fireclay brick, which is 1 m by 1 m on a side. During steady-state operation is such that three surfaces of the column are maintained at 500 K while the remaining surface is exposed to 300 K. Using a grid of ∆x=∆y=0.25 m, determine the two-dimensional temperature distribution in the column.
(1,3)
(2,3)
(3,3)
(1,2)
(2,2)
(3,2)
(1,1)
(2,1)
(3,1)
Ts=300 K

68. System of Linear Equations-4 1
-800
-500
-1000
-300 =
System of Linear Equations

69. Matrix Inversion Method

70. Iteration Method using Excel

71. Jacobi Iteration Method using Excel

72. Error Iteration Method using Excel

73. Gauss-Seidel Iteration Method using Excel

74. Error Iteration Method using Excel

78. Iteration Method using Excel