Presenting Error and Convergence
In the AMME2000 and AMME2960 units of study you will be asked to present the error and
convergence rates for different numerical methods to demonstrate the accuracy and performance
of the methods. This section will show how to calculate and present the error and convergence. The
example problem used is calculation of the gradient of the function
f
(
x
) =
e
x
using the backward
and central difference schemes. This problem is the second task from the second week’s tutorial.
To refresh, the backward difference scheme is:
f
0
(
x
)
≈
f
(
x
)
-
f
(
x
-
Δ
x
)
Δ
x
(1)
and the central difference scheme is:
f
0
(
x
)
≈
f
(
x
+ Δ
x
)
-
f
(
x
-
Δ
x
)
2Δ
x
(2)
There are two ways that the error from these schemes could be calculated. The first is by performing
a Taylor series expansion of the terms on the right-hand side and rearranging to isolate the error
term. The second is by evaluating the two numerical schemes to find the approximate gradient
of a function and then calculating the error from the difference between the true gradient of the
function and the approximate gradient. The first method will give a mathematical expression for
the error when the scheme is applied to any function and the second will give a numerical value
for the error when the scheme is applied to a particular function at a particular point or over a
particular domain. Both measures of error are valid and should be used where appropriate.
You should use the Taylor series method if you are asked to:
•
Find or derive an expression for the error
•
Analytically obtain the order of accuracy of a scheme
•
Explain why or prove that a scheme possesses a certain order of accuracy or error
Note that the Taylor series is the appropriate method for finding an analytical expression for the
error from a finite difference scheme such as the backward or central schemes.
Other numeri-
cal methods may require different techniques to analytically demonstrate convergence and error
properties.
You should calculate the error by evaluating the numerical scheme for a particular problem if you
are asked to:
•
Calculate the value of the error
•
Perform a convergence study
Here we will look at calculating the value of the error and performing a convergence study. Watch
the videos included with the material for the first week for the use of the Taylor series to find an
analytical expression for the error.
Let’s begin by calculating the magnitudes of the errors from the backward and central differences
schemes at the point
x
= 0
.
5.
We’ll express the error as a percentage of the magnitude of the
analytical gradient:
1
=
|
f
0
(
x
)
-
e
f
0
(
x
)
|
|
f
0
(
x
)
|
×
100
(3)
In this equation
f
0
is the analytical gradient and
e
f
0
is the numerical gradient. The subscripts
b
and
c
will be used for errors and gradients from the backward and central schemes respectively,
e.g.
