Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 175

5.2.2 First Derivative

Discretization of the convective term in Navier-Stokes equation
requires the approximation of the irst derivative ∂ ( ρu G ) /∂ x . We
shall now describe some approaches to the approximation of the
irst derivative of any quality. In the previous section, one means
of deriving approximations to the irst derivative was presented.
There are more systematic approaches that are better suited to the
derivation of more accurate approximations; some of these will be
described later.

5.2.3 Taylor Series Expansion

Any continuous differentiable function G ( x ) can, in the vicinity of x 1 ,
be expressed as a Taylor series:

¥ µ

22

¥µ u G

(- ) xx

u G

G G

i

u § ·

() x ( ) ( - ) x x x i i

¦¶ ¦ ¶ i i

§·

u

2

x 2! x

33

¥ µ

nn

¥ µ

u G u G

(- ) xx (- ) xx

i i

(5.47)

i

–––

i H O T ..

¦ ¶ ¦ ¶

3 n

3!

u

n !

u

§ ·

x

§ ·

x

where H.O.T means “higher order terms.” By replacing x with x i +1 or
x i- 1 in this equation, one obtains expressions for the variable values
at these points in terms of the variable and its derivatives at x i . This
can be extended to any other point near x i in terms of the function
values at neighboring points. For example, using Eq. (5.47) for G at
x i +1 , we can show that

¥ µ

2

¥µ u G GG

- x - x

u G

i 1 i i 1 i

-

¦¶ ¦ ¶

§·

2

u

x x - x 2

u § ·

x

i i 1 i

i

(5.48)

23

¥ µ

u G

( x -) x

i 1 i

- HOT . .

¦ ¶
u § ·

3

6

x

i

Another expression may be derived using the series expression
(5.47) at x i -1 :

¥ µ

GG

- xx -

2

u G u G

¥µ i i -1 i i -1

-

¦¶

¦ ¶

§· u

2

x x - x 2

u § ·

x

i i i -1

i

(5.49)

2

¥ µ

3

(- xx )

u G

i i

-1

- HOT . .

¦ ¶
u § ·

3

6

x

i