Parabolic Partial Differential Equations - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Parabolic Partial Differential

Equations

Defining Parabolic PDE’s

• The general form for a second order linear PDE with two independent variables and

one dependent variable is

• Recall the criteria for an equation of this type to be considered parabolic
• For example, examine the heat-conduction equation given by

Then

thus allowing us to classify this equation as parabolic.

2 0

2 2 2

2 A ^ xu  B  x  uy  C  yu  D 

B^2  4 AC  0

, where

0

(^2404) ( )( 0 ) 

B  AC   

t

T x

T    

 2

2  A ^ ,^ B ^0 , C ^0 , D ^1

Discretizing the Parabolic PDE

Schematic diagram showing interior nodes

x

i  1 i i  1

 x  x

For a rod of length divided into nodes

The time is similarly broken into time steps of

Hence corresponds to the temperature at node ,that is,

and time

L n  1  x  nL  t Tij i

x   i  x  t   j  t 

The Explicit Method

If we define we can then write the finite central divided difference

approximation of the left hand side at a general interior node ( ) as

where ( ) is the node number along the time.

n  x  L i

x

i  1 i i  1

 x  x

 ^2

(^2 ) x

T T T x

T (^) ij ij ij i j 

   

 (^)  

j

The Explicit Method

Substituting these approximations into the governing equation yields

Solving for the temp at the time node gives

choosing,

we can write the equation as,

  t

T T x

Ti j Tij Tij ij ij    

 ^   ^1 1 2 1 ^2

j  1

Ti j ^1  Tij  (  xt ) 2  T i  j 1  2 Tij  Ti  j 1 

( x )^2

t    

Ti j ^1  Ti j   T^ i  j 1  2 T i j  Ti  j 1 

The Explicit Method

• This equation can be solved explicitly because it can be written for each internal

location node of the rod for time node in terms of the temperature at time node

• In other words, if we know the temperature at node , and the boundary

temperatures, we can find the temperature at the next time step.

• We continue the process by first finding the temperature at all nodes , and

using these to find the temperature at the next time node,. This process

continues until we reach the time at which we are interested in finding the

temperature.

Ti j ^1  Ti j   T^ i  j 1  2 T i j  Ti  j 1 

j  1

j

j  0

j  1 j  2

Example 1: Explicit Method

Recall,

therefore,

Then,

C

k   

7800 490

54     1. 4129  10 ^5 m^2 / s

 (^) x ^2

t

 ^2

5

1. 01

 1. 4129  10 ^3

 0. 4239.

Number of time steps,

Boundary Conditions

All internal nodes are at

for This can be represented

as,

t

t final tinitial

^9 ^0

(^10025) forall 0 , 1 , 2 , 3 5

(^0)  

  

  j T C

T C j

j

20  C

t  0 sec^.

Ti^0  20  C ,forall i 1,2,3,

Example 1: Explicit Method

Nodal temperatures when , :

We can now calculate the temperature at each node explicitly using the

equation formulated earlier,

t  0 sec

T 0^0  100  C

Interiornodes 20

40

30

20

10



T C

T C

T C

T C

T 5^0  25  C

j  0

Ti j ^1  Ti j   T i  j 1  2 T i j  Ti  j 1 

Example 1: Explicit Method

Nodal temperatures when (Example Calculations)

setting ,

Nodal temperatures when , :

t  6 sec

T 02  100  C BoundaryCondition

Interiornodes

22. 442

42

32

22

12



T C

T C

T C

T C

T 52  25  C Boundary Condition

i  0 T 02  100  C BoundaryCondition i  (^1) i  2

t  6 sec j  2

      C

T T T T T

 

 

 

   

   

59. 073

60. 912 5. 1614

61. 912 0. 423912. 176

62. 912 0. 423920 2 ( 53. 912 ) 100 12 11  21 2 11 01 ^      C

T T T T T

 

 

 

   

   

34. 375

20 14. 375

20 0. 423933. 912

20 0. 423920 2 ( 20 ) 53. 912 22 21  31 2 21 11

j  1

Example 1: Explicit Method

Nodal temperatures when (Example Calculations)

setting ,

Nodal temperatures when , :

t  9 sec

T 03  100  C BoundaryCondition

Interiornodes

22. 872

43

33

23

13



T C

T C

T C

T C

T 53  25  C BoundaryCondition

j  2

i  0 T 03  100  C BoundaryCondition i  1 i ^2

t  9 sec j ^3

      C

T T T T T

 

 

 

   

   

65. 953

66. 073 6. 8795

67. 073 0. 423916. 229

68. 073 0. 423934. 375 2 ( 59. 073 ) 100 13 12  22 2 12 02 ^      C

T T T T T

 

 

 

   

   

39. 132

40. 375 4. 7570

41. 375 0. 423911. 222

42. 375 0. 423920. 899 2 ( 34. 375 ) 59. 073 23 22  32 2 22 12

The Implicit Method

WHY:

• Using the explicit method, we were able to find the temperature at each

node, one equation at a time.

• However, the temperature at a specific node was only dependent on the

temperature of the neighboring nodes from the previous time step. This is

contrary to what we expect from the physical problem.

• The implicit method allows us to solve this and other problems by

developing a system of simultaneous linear equations for the temperature

at all interior nodes at a particular time.

The Implicit Method

The second derivative on the left hand side of the equation is

approximated by the CDD scheme at time level at node ( ) as j  1

t

T x

T 

  

 2

2 

 ^2

(^2 ) x

T T T x

T (^) ij ij ij i j 

   

 (^)   

i

The Implicit Method

Substituting these approximations into the heat conduction equation yields

t

T x

T 

  

 2

2 

  t

T T x

Ti j Ti j Ti j ij i j 

  

 ^    ^1 2

1 1

1 1 ^1

From the previous slide,

Rearranging yields

given that,

The rearranged equation can be written for every node during each time step.

These equations can then be solved as a simultaneous system of linear equations

to find the nodal temperatures at a particular time.

The Implicit Method

  t

T T x

Ti j Tij Tij ij ij    

 ^    ^1 2 1 1 2 1 11 

  Ti (^)  j 1  1 ( 1  2 ) Tij ^1   Ti  j 1 ^1  Tij

 x ^2

t 

  