Geoscience Reference
In-Depth Information
boundary conditions (4.2); however, for complex problems, this may be difficult, and
one may use the values of
p
at boundaries as boundary conditions for solving Eq. (4.8).
Because Eqs. (4.1) and (4.8) have the same formulation except for the source terms,
the same numerical method can be used to solve Eq. (4.8) and find the approximate
solution of
p
. The above numerical accuracy analysis can then be conducted.
Consistency
The system of discretized equations is considered to be consistent with the original
differential equation if it is equivalent to the differential equation at each grid point
when the grid spacing reduces to zero.
The consistency analysis can be conducted by expanding all nodal values in the dis-
cretized equations as Taylor series about a single point. For consistency, the obtained
expression should be made up of the original partial differential equation and a remain-
der, and the remainder should reduce to zero at each grid point as the grid spacing
reduces to zero.
Stability
Numerical stability is concerned with growth or decay in errors introduced at any
stage of the computational process. In practice, because of limited computer stor-
age, an infinite decimal number is truncated to a finite number of significant figures,
thereby introducing round-off errors. A numerical algorithm is said to be stable if the
cumulative effect of the errors produced during its application is negligible.
The von Neumann and matrix methods are commonly used for stability analysis
(see Hirsch, 1988; Fletcher, 1991). Both methods can predict whether there will be a
growth in numerical errors including the round-off contamination between the true
solution of the numerical algorithm and the actually computed solution.
Convergence
A solution of the discretized algebraic equations is said to be convergent if the approx-
imate solution approaches the exact solution of the original differential equation for
each dependent variable as the grid spacing reduces to zero. Thus, for problem (4.1),
convergence requires
f
0.
Proving the convergence of a numerical algorithm is generally very difficult, even
for the simplest cases. Nevertheless, for a restricted class of problems, convergence
can be established via the Lax equivalent theorem, which was described as follows
(Richtmyer and Morton, 1967; Fletcher, 1991):
→
f
,as
x
→
Given a properly posed linear initial value problem and a finite difference approx-
imation to it that satisfies the consistency condition, stability is the necessary and
sufficient condition for convergence.
The Lax equivalent theorem is very useful to show the convergence through
the stability and consistency analyses, which are much easier. However, most of
