Finite difference method

1. FINITE DIFFERENCE
In numerical analysis, two different approaches are commonly used:
The finite difference and the finite element methods. In heat transfer
problems, the finite difference method is used more often and will be
discussed here. The finite difference method involves:
Establish nodal networks
Derive finite difference approximations for the governing
equation at both interior and exterior nodal points
Develop a system of simultaneous algebraic nodal
equations
Solve the system of equations using numerical schemes

2. The Nodal Networks

3. Finite Difference Approximation

4. Finite Difference Approximation cont.

5. Finite Difference Approximation cont.

6. A System of Algebraic Equations

7. Matrix Form

8. Numerical Solutions

9. Iteration

10. Example

11. Example (cont.)

12. Example (cont.)

13. Summary of nodal finite-difference
relations for various configurations:
Case 1 Interior Node
0
4T
T
T
T
T n
m,
n
1,
m
n
1,
m
1
n
m,
1
n
m, 



 




14. Case 2
Node at an internal corner with convection
1, , 1 1, , 1 ,
2( ) ( ) 2 2(3 ) 0
m n m n m n m n m n
h x h x
T T T T T T
k k
    
 
      

15. Case 3
Node at a plane surface with convection
1, , 1 , 1 ,
(2 ) 2 2( 2) 0
m n m n m n m n
h x h x
T T T T T
k k
   
 
     

16. Case 4
Node at an external corner with convection
, 1 1, ,
( ) 2 2( 1) 0
m n m n m n
h x h x
T T T T
k k
  
 
    

17. Case 5
Node at a plane surface with uniform heat flux
0
4
'
'
2
)
2
( ,
1
,
1
,
,
1 




 

 n
m
n
m
n
m
n
m T
k
x
q
T
T
T