K. M. Isaac Professor of Aerospace Engineeringweb.mst.edu/~isaac/aeme339.html/fs01_files.html/topic3_PDE.pdf · 29-08-2001 · Computational Fluid Dynamics (AE/ME 339) K. M. Isaac

8/29/01 Topic 3 Discretization of PDE 1

AE/ME 339

K. M. IsaacProfessor of Aerospace Engineering


Computational Fluid Dynamics (AE/ME 339) K. M. IsaacMAEEM Dept., UMR

Dicretization of Partial Differential Equations (CLW: 7.2, 7.3)

We will follow a procedure similar to the one used in the previous classWe consider the unsteady vorticity transport equation, noting that the equation is non-linear.



Vorticity vector:

ux

i

�

�

�

vy

j

�

�

�

wz

k

�

�

�

_ _ _curl v v� � ��

Is a measure of rotational effects.

�� 2� �where is the local angular velocity of a fluid

element.

=



For 2-D incompressible flow, the vorticity transport equation is given by

�� 2

��

��

�

��

�

� vy

vx

ut

(1)

2

2

2

22

yx �

��

�

��

��

� - kinematic viscosity ��

��

��

�

sm2



As in the case of ODE ,the partial derivatives can be discretizedUsing Taylor series

� � � � � �

� �

� �� n

n

Ryxuy

kx

hn

yxuy

kx

h

yxuy

kx

hyxukyhxu

��

��

�

�

��

�

�

��

��

�

�

��

�

�

��

��

�

�

��

�

��

�

,!1

1

..................................,!2

1

,,,

1

2

(2)



( ) nnR O h k� ��

� �(3)

We can expand in Taylor series for the 8 neighboring points of(i,j) using (i,j) as the central point.



� � � �xxxxxxjiji uxuxxuuu

32

,,1�

��

��

� � � �xxxxxxjiji uxuxxuuu

32

,,1�

��

��

(5)

(4)



2

2

,xuu

xuu xxx

�

��

�

��Here etc.

Note: all derivatives are evaluated at (i,j)

Rearranging the equations yield the following finite differenceformulas for the derivatives at (i,j).

� �1 , ,i j i ju uu O xx x

��

� � ��

(6)



� �, 1,i j i ju uu O xx x

�

��

� �

(7)

� �21, 1,

2i j i ju uu O x

x x� �

��

(8)

� ��

221, , 1,

22

2i j i j i ju u uu O xx x

� ��

� � �(9)



Eq.(6) is known as the forward difference formula.

Eq.(7) is known as the backward difference formula.

Eq.(8) and (9) are known as central difference formulas.

Compact notation:

1 1, ,2 2

,

i j i j

x i j

u uu

x�

� �

�

��

(10)



The Heat conduction problem (ID)

x x

T Tk k xx x x

� � ��

� � � �xx

Tkx

��

�

x x+�x

Consider unit area in the direction normal to x.Energy balance for a CV of cross section of area 1 and length �x:

Volume of CV, dV = 1 �x



Change in temperature during time interval �t, = �T

Increase in energy of CV :

1 pTx c t HOTt

��

� � � � ��

This should be equal to the net heat transfer across the two faces

x x x

T T Tk t k k x t HOTx x x x

� ��

� � � � ��



Equating the two and canceling �t �x gives

pT Tc kt x x

��

� � ��

Note: higher order tems (HOT) have been dropped.If we assume k=constant, we get

2

2pT Tc kt x

��

��



2 2

2 2p

T k T Tt c x x

��

� � ��

� � �Or

p

kc

��

� is the thermal diffusivity.Where

Letting � = x/L, and � = �t/L2, the above equation becomes

2

2

T T� �

� ��

� �



The above is a Parabolic Partial Differential Equation.

2

2

xu

tu

�

��

�

�(11)



Physical problem

A rod insulated on the sides with a given temperature distributionat time t = 0 .Rod ends are maintained at specified temperature at all time.Solution u(x,t) will provide temperature distribution along the rodAt any time t > 0.

2

2

xu

tu

�

��

�

�10,0 ttlx ��



IC:

)()0,( xfxu � lx ��0 (12)

BC:

)(),()(),0(

1

0

tgtlutgtu

�

� 10 tt ��

10 tt ��(13)



Difference Equation

Solution involves establishing a network of Grid points as shown in the figure in the next slide.

Grid spacing: ,Mlx ��

Ntt 1

��




M,N are integer values chosen based on required accuracyand available computational resources.

Explicit form of the difference equation

� �2,1,,1,1, 2

xuuu

tuu ninininini

�

��

�

�� (14)

Define 2( )t

x�

��

�



Then

� �, 1 1, , 1,1 2i n i n i n i nu u u u� � �� (15)

Circles indicate grid points involved in space differenceCrosses indicate grid points involved in time difference.



Note:At time t=0 all values � �ii xfu �0, are known (IC).

In eq.(15) if all niu , are known at time level tn, , 1i nu�

can be calculated explicitly.

Thus all the values at a time level (n+1) must be calculated beforeadvancing to the next time level.

Note: If all IC and BC do not match at (0,0) and )0,(l , it should behandled in the numerical procedure.Select one or the other for the numerical calculation.There will be a small error present because of this inconsistency.



Convergence of Explicit Form.

Remember that the finite difference form is an approximation.The solution also will be an approximation.

The error introduced due to only a finite number of terms in theTaylor series is known as truncation error, ��

The solution is said to converge if

0� � when , 0x t� � �

Error is also introduced because variables are represented by a finite number of digits in the computer.This is known as round-off error.



as

Therefore, the solution converges.

2[( ) ]O x� � �

0� � 0x� �

From the above

For the explicit method, the truncation error, � is

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K. M. Isaac Professor of Aerospace Engineeringweb.mst.edu/~isaac/aeme339.html/fs01_files.html/topic3_PDE.pdf · 29-08-2001 · Computational Fluid Dynamics (AE/ME 339) K. M. Isaac