Numerical Simulation for Blood Flow - Microsurgery: Advances, Simulations, and Applications - page 176

Still another expression may be obtained by using Eq. (5. 47) at
both x i -1 and x i +1 :

2 2

¥ µ

2

¥µ u G GG

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

u G

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

-

u§ ·

¦¶ ¦ ¶

§·

2

u

x x - x 2( x - x )

x

i i -1 i i -1

i

i

3 3

¥ µ

3

( x -) x ( x -) x

u G

i 1 i i i -1

- HOT . . (5.50)

¦ ¶
u§ ·

3

6( x - x )

x i 1 i -1

i

Equations (5.48-5.50) are exact if all terms on the right-hand
side are retained. Because the higher-order derivatives are unknown,
these expressions are not of great value as they stand. However, if
the distance between the grid points i.e., x i - x i- 1 and x i +1 - x i are
small, the higher-order terms are locally very large. Ignoring the
latter possibility, approximations to the irst derivative result from
truncating each of the series after the irst terms on the right-hand
sides:

u ¥µ z G GG

-
-

i 1

1

i

(5.51)

¦ §·

u

x x x

i i i

u ¥µ z G GG

-
-

i i -1

-1

(5.52)

¦ §·

u

x x x

i i i

u ¥µ z G GG

-
-

i 1 i -1

(5.53)

¦ §·

u

x x x

i i 1 i -1

These are the forward- (FDS), backward- (BDS), and central-
difference (CDS) schemes, respectively. The terms that were deleted
from the right-hand sides are called the truncation errors; they
measure the accuracy of the approximation and determine the rate
at which the error truncated term is usually the principal source of
error. The truncation error is the sum of products of a power of the
spacing between the points and higher order derivatives at the point
x = x i :
FH E B

m m 1 n

E B ––– E B

2 1 (5.54)
where δx is the spacing between the points (assumed all equal for
the present) and the α's are higher-order derivatives multiplied by
constant factors. From Eq. (5.54), we see that the terms containing
higher powers of δx are smaller for small spacing so that the leading
term (the one with the smaller exponent) is the dominant one. As δx is
reduced, the above approximations converge to the exact derivatives
with an error proportional to ( δx ) m , where m is the exponent of

() x () x () x

m 1 m n