38
2. BasicFinite-Difference Methods
From the definition of 8 nx ,
821 = 8 (8 I) = I(x + ßX) - 2/(x) + I(x - ßx)
x
x x
(ßx)2
'
and a conventional Taylor series analysis of the truncation error shows that
It foltows that 82x8;1 is a second-order approximation to the third derivative of
I, since
Substitution of the preceding into (2.8) yields
(2.9)
Expansion of this formula via the operator definition (2.7) yields the centered
fourth-order difference (2.6). Although it allows finite-difference equations to be
expressed in a very compact form, operator notation will not be used for alt finitedifference equations throughout the remainder of this book , but will be reserved
for complicated formulae that become unwieldy when written in expanded form .
Most of the finite-difference schemes considered in the remainder of this chapter
are sufficiently simple that they will be expressed without using operator notation.
We now turn from the consideration of individual finite differences to examine
the accuracy of an entire finite-difference scheme. Suppose that an approximation
to the advection equation
al/J
al/J
-+c-=o
(2.10)
at
ax
is to be obtained at the grid points (n St , j tsx) , where n and j are integers. It
is convenient to represent the numerical approximation to l/J«nßt, jßx) in the
shorthand notation cl/}. One possible finite-difference formula for the numerical
approximation of (2.10) is
(2.11)
when c > O. this is known as the "upstream" or "donor-cell" scheme . The accuracy of a finite-difference scheme is characterized by the residual error with which