Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 178

¥µ ¥µ
¦¶ ¦¶

u G u G

-

§· §· u u

x x

1 1

¥ µ

2

u G

i i -

2 2

z

¦ ¶
u § ·

2

1
2

x

( x -) x

i

i 1 i -1

G

(- x x )

G

( x -)-( x

G

x - x

i 1 i i -1 i -1 i 1 i i i 1 i -1

z

(5.58)

1
2

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

i 1 i -1 i 1 i i i -1

For equidistant spacing of the points, expression (5.56) becomes
simple:

¥ µ

2

G G G

-2
()

u G

i 1 i -1

z

(5.59)

¦ ¶
u§ ·

2 2

x

E

x

i

Higher-order approximations for the second derivative can be
obtained by including more data points, say x i -2 or x i +2 .
Finally, one can use interpolation to it a polynomial degree n
through n 1 data points. From that interpolation, approximations

to all derivatives up to the n th can be obtained by differentiation.

+

T

N

E

W P

S

B

Figure 5.10 Example of computational grid in 3D.