50
3. Finite Difference Methods
derivative twice. This is the only approach possible when the fluid properties
are variable, since we need the derivative of the product of diffusion coefficient and the first derivative. We next consider approximations to the second
derivative; application to the diffusive term in the conservation equation will
be discussed later.
Geometrically, the second derivative is the slope of the line tangent to the
curve representing the first derivative, see Fig. 3.2. By inserting approximations for the first derivatives at locations xi+l and xi, an approximation for
the second derivative is obtained:
All such approximations involve data from at least three points.
In the above equation, the outer derivative was estimated by FDS. For
inner derivatives one may use a different approximation, e.g. BDS; this results
in the following expression:
One could also use the CDS approach which requires the first derivative at
at xi-1 and xi+l. A better choice is to evaluate @/ax at points halfway
between xi and xi+l and xi and xi-1. The CDS approximations for these
first derivatives are:
respectively. The resulting expression for the second derivative is:
For equidistant spacing of the points, expressions (3.28) and (3.30) become:
Taylor series expansion offers another way of deriving approximations to
the second derivative. Using the series at xi-1 and xi+l given above, one can
re-derive Eq. (3.28) with an explicit expression for the error: