1. Home assignment #3 (1) (Total 3 problems) Due: 12 November 2018
Consider the following 1D heat conduction problem (Hint: Case Study in 9.5) 1 2 3
N-3
N-2
N-1 N
- Governing equation
(1) Using the governing equation and boundary conditions given, derive the exact solution. (Hint: Linear second-order ordinary differential equation)
(2) Using the Finite Difference formula, derive the difference equations, system of linear equations and corresponding matrix equation.

2. Home assignment #3 (2)
(3) Solve the matrix equation with the following information - N = 10, 50, h’ = 0.1 m-2, Ta = 100 ℃, T(0) = 0 ℃, T(5)=30 ℃
Use the following numerical methods (a) Gauss-elimination (Naïve) (b) Gauss-elimination (Pivoting) (c) LU decomposition (d) Cholesky factorization (e) Gauss-Seidel iteration (f) Gauss-Seidel iteration with over-relaxation (Use your own λ) (g) Gauss-Seidel iteration with under-relaxation (Use your own λ) (h) Jacobi iteration
(Note) When summarizing the results, - Draw (T-x) graph, and compare the numerical results with exact solution.
(4) Using results from (3) - Discuss the variation of numerical error with N - For the iteration methods (e, f, g, h) : Draw (relative error-iteration) graphs : Discuss the convergence speed for various methods : choose your own stopping criterion

3. HW#3-1: Solution
(1) Using the governing equation and boundary conditions given, derive the exact solution. (Hint: Linear second-order ordinary differential equation)
- Governing equation
- General solution
- (Characteristic equation)
- (General solution)
- Particular solution
- Undetermined coefficient
h’ = 0.1 m-2, Ta = 100 ℃
T(0) = 0 ℃, T(5)=30 ℃

4. HW#3-1: Solution
(2) Using the Finite Difference formula, derive the difference equations, system of linear equations and corresponding matrix equation.
Using finite difference scheme
(Finite difference equation)
System of linear equations
In matrix form

5. HW#3-1: Solution
(3) Results: (a-d) Direct methods, (e-h) Iterative methods.
N=10
N=50
N=100
Stopping criteria = 1.e-5%, Max iteration = 30000

6. HW#3-1: Solution (4) Discussion (e)-(h) Iterative methods (N=100)
(a)-(d) Direct methods
True Error
N=10
N=50
N=100
Gauss Naïve
3.4592E-02
1.3901E-03
3.4755E-04
Gauss Pivot
LU Decomposition
Cholsky
The true error becomes smaller as N increases.
When Δx becomes 1/10 (N=10N=100), it’s error becomes smaller by 1/100.
In this problem, the Gauss-Seidel with over-relaxation shows the fastest convergence speed.

7. Home assignment #3 (3) (Problem 2) (Problem 3)
Solve the problem in the textbook (P12.11)
(Problem 3)
Solve (Problem 2), but increase the number of nodes to (50x50) and use the boundary condition shown below. Draw the temperature distribution on the plane.
T = 100 ℃
L = 1 m
T = 0 ℃
T = 0 ℃
H = 1 m
T = 0 ℃

8. HW#3-2 & #3-3: Solution
Theory

9. HW#3-2 & #3-3: Solution
Script and programs
#3.2
#3.3

10. HW#3-2 & #3-3: Solution (Problem 2) (Problem 3)
Stopping criteria = 1.e-5%, Max iteration = 3000
(Problem 2)
(Problem 3)
Contour plot for (50x50)
T12 = 56.25
T11 = 68.75
T22 = 31.25
T21 = 43.75