Page 273 - Numerical Methods for Chemical Engineering

262 6 Boundary value problems

be satisfied locally:
2 2
∂ ϕ ∂ ϕ
− − = f (x i , y j ) (6.17)
∂x 2 (x i ,y j ) ∂y 2 (x i ,y j )
The key idea of finite differences is to approximate these local derivatives by algebraic
differences of the field values at neighboring grid points.

Finite difference approximations

In the finite difference method, we approximate the value of the derivative of f (x)at x 0
using a truncated Taylor series approximation for small x,

df 2
f (x 0 + x) = f (x 0 ) + ( x) + O[( x) ] (6.18)
dx
x 0
to yield

f (x 0 + x) − f (x 0 )
df
= + O[ x] (6.19)
dx x
x 0
Similarly, from the expansion in the opposite direction,

df 2
f (x 0 − x) = f (x 0 ) + (− x) + O[( x) ] (6.20)
dx
x 0
we obtain
f (x 0 ) − f (x 0 − x)

df
= + O[ x] (6.21)
dx x
x 0
Both of these one-sided difference approximations have errors proportional to x, the
spacing between successive grid points. Subtracting the two,
(
2 2

df ( x) d f 3
f (x 0 + x) = f (x 0 ) + ( x) + + O[( x) ]
dx 2 dx 2
x 0 x 0
(
2 2

df ( x) d f 3
− f (x 0 − x) = f (x 0 ) − ( x) + + O[( x) ] (6.22)
dx 2 dx 2
x 0 x 0
the zeroth-and second-order terms cancel out, and the resulting central difference approxi-
mation is second-order accurate,

f (x 0 + x) − f (x 0 − x)
df 2
= + O[( x) ] (6.23)
dx 2( x)
x 0
The error of (6.23) is proportional to the square of the grid point spacing, thus it decays
to zero as x → 0 more rapidly than the errors of one-sided formulas. Equation (6.23) is
more accurate than (6.19) or (6.21).