Geoscience Reference
In-Depth Information
Figure 4.1
Example of numerical solution.
where
f
is the approximate solution of
f
, which is subject to boundary conditions:
f
1
f
N
=
f
a
,
=
f
b
(4.4)
where
f
1
and
f
N
are the values of
f
at
x
x
1
and
x
N
, respectively.
The system of algebraic equations consisting of discrete equations (4.3) and bound-
ary conditions (4.4) is used to determine the approximate solution (
f
1
,
f
2
,
=
,
f
N
)on
the computational grid. A direct or iterative solution method may be adopted to solve
the algebraic equations. The obtained approximate solution is a discrete function,
which is shown as solid circles in Fig. 4.1.
The quality of the approximate solution usually relies on the computational grid
used, the discretization method for the governing equation, and the solution method
for the discretized equations.
...
4.1.2 Properties of numerical solution
The most important properties of numerical solution are accuracy, consistency,
stability, and convergence. A brief overview of these terms is given below. Complete
descriptions can be found in Hirsch (1988), Fletcher (1991), etc.
Accuracy
Numerical accuracy refers to how well a discretized equation approximates to the
differential equation. Eq. (4.3) is said to have an accuracy of
m
th-order of
x
, if the
x
m
:
residual (error) is proportional to
L
d
(
f
;
x
i
)
=
x
m
R
L
=
(
f
;
x
i
)
−
(
)
L
O
(4.5)
The residual term on the right-hand side of Eq. (4.5) can be obtained with the Taylor
series expansion method. However, it is usually difficult to judge the overall accuracy
