SYLLABUS Previous: 1.4 Numerical discretization
Up: 1.4 Numerical discretization
Next: 1.4.2 Sampling on a
1.4.1 Convergence, Richardson extrapolation
One of the most important properties of a numerical approximation based on
a finite number of data points is that the approximation converges to
the exact value as more information is gathered (exercise 1.01).
This convergence can be defined locally for any arbitrary point in
space and time (more restrictive) or by monitoring a global quantity
(more permissive).
The convergence rate can be estimated experimentally from a geometric
sequence of approximations
, by successively refining the
numerical resolution
, for example by doubling the
resolution a couple of times when

(1.4.1#eq.1) 
Knowing the convergence rate
, it is possible to further improve
the accuracy of the numerical approximation by using a socalled
Richardson extrapolation

(1.4.1#eq.2) 
where a more accurate solution
is obtained from an evaluation
and its refinement
.
Let's take an example and approximate the function
on a
grid in
, simply by sampling the value of the function in the
middle of
intervals at
.
The approximation for the value at the origin
converges to zero provided that
, which is best visualized in a convergence study using
a linlin plot (figure 1.4.1#fig.1, left).
Figure 1.4.1#fig.1:
Convergence studies for the approximation
showing the same data in a linlin plot (left)
and a loglog plot (right).

The convergence rate for the approximation measured at the origin
drops as
as shown by the lower slope in the
loglog plot (figure 1.4.1#fig.1, right).
Not apparent in the loglog plot is that the approximation
converges to a different value
when
.
The midpoint rule (sect.3.3) can be used to approximate an
integral by computing the area of small boxes in order to evaluate the
global quantity
with
. Global convergence is achieved for
;
because of the weak singularity, the convergence rate drops from the
expected from a quadrature of smooth
functions to
when
.
SYLLABUS Previous: 1.4 Numerical discretization
Up: 1.4 Numerical discretization
Next: 1.4.2 Sampling on a
