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3. Finite Difference Methods
In the following two sections, only the one dimensional case is considered.
The coordinate may be either Cartesian or curvilinear, the difference is of little importance here. In multidimensional finite differences, each coordinate is
usually treated separately so the methods developed here are readily adapted
to higher dimensionality.
3.3 Approximation of the First Derivative
Discretization of the convective term in Eq. (3.1) requires the approximation
of the first derivative, d(pu4)ldx. We shall now describe some approaches to
approximation of the first derivative of a generic variable 4; the methods can
be applied to the first derivative of any quantity.
In the previous section, one means of deriving approximations to the
first derivative was presented. There are more systematic approaches that
are better suited to the derivation of more accurate approximations; some of
these will be described later.
3.3.1 Taylor Series Expansion
Any continuous differentiable function 4(x) can, in the vicinity of xi, be
expressed as a Taylor series:
(x - xi)2 d 2 4
4(x) = +(xi) + (2 - xi)
(a,.) +
i
where H means "higher order terms". By replacing x by xi+l or xipl in this
equation, one obtains expressions for the variable values a t these points in
terms of the variable and its derivatives a t xi. This can be extended to any
other point near xi, for example, xi+2 and xi-2.
Using these expansions, one can obtain approximate expressions for the
first and higher derivatives at point xi in terms of the function values a t
neighboring points. For example, using Eq. (3.3) for 4 a t xi+l, we can show
that:
Another expression may be derived using the series expression (3.3) a t xi-1:
3. Finite Difference Methods
In the following two sections, only the one dimensional case is considered.
The coordinate may be either Cartesian or curvilinear, the difference is of little importance here. In multidimensional finite differences, each coordinate is
usually treated separately so the methods developed here are readily adapted
to higher dimensionality.
3.3 Approximation of the First Derivative
Discretization of the convective term in Eq. (3.1) requires the approximation
of the first derivative, d(pu4)ldx. We shall now describe some approaches to
approximation of the first derivative of a generic variable 4; the methods can
be applied to the first derivative of any quantity.
In the previous section, one means of deriving approximations to the
first derivative was presented. There are more systematic approaches that
are better suited to the derivation of more accurate approximations; some of
these will be described later.
3.3.1 Taylor Series Expansion
Any continuous differentiable function 4(x) can, in the vicinity of xi, be
expressed as a Taylor series:
(x - xi)2 d 2 4
4(x) = +(xi) + (2 - xi)
(a,.) +
i
where H means "higher order terms". By replacing x by xi+l or xipl in this
equation, one obtains expressions for the variable values a t these points in
terms of the variable and its derivatives a t xi. This can be extended to any
other point near xi, for example, xi+2 and xi-2.
Using these expansions, one can obtain approximate expressions for the
first and higher derivatives at point xi in terms of the function values a t
neighboring points. For example, using Eq. (3.3) for 4 a t xi+l, we can show
that:
Another expression may be derived using the series expression (3.3) a t xi-1: