Finite Difference Method for Solving Partial Differential Equations, Lecture notes of Metallurgy

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Finite Difference Method

Partial difference equation

































  





z

c

y

c

x

c

D c D

t

c

subject to boundary conditions

q   D  c  q

c  c

starting from initial values

i

c ( x , y , z , 0 )  c

y

x

We want to find c(x,y,z,t)

A

B C

D

3

3 3

2

2 2

 

t  k  t t  k  t t t  k  t

t c

t

t c

t

c c k t t c k t t

3

3 3

2

2 2

 

t  k  t t  k  t t t  k  t

t c

t

t c

t

c c k t t c k t t

.....

2

( ) ( ) 



     





  t

c k t t c k t t

t

c

t k t

.....

( ) ( ) 



    





  t

c k t t c k t

t

c

t k t

.....

( ) ( ) 



    





  t

c k t c k t t

t

c

t k t

Forming discrete equation

( 2 ) 1 1

1 k

i

k

i

k

i

k

i

k

i

c c c c c  

    

2 x

D t





What happens at the boundary?

i=0 i=1^ i=2^ i=

k k k k k

c c c c c

when the flux at the boundary is zero, then

k k c c 1 1

 