The trapezoidal rule is a simple average of the forward-Euler and
backward-Euler schemes. It can be shown that the local truncation
error scales with .

(1.6)

For ordinary differential equations, the trapezoidal rule is an
application of the method, which itself is a special case of a
second-order Runge-Kutta method. For more details see
[6].

Figure 1.3:
Graphical illustration of
the trapezoidal method. Starting at point 1, we get point 2 by
taking the derivatives at point 1 and point 2, and extrapolating
their average in point 1.

Table 1.1:
The numerical schemes
of interest. Not all of them are mentioned in the
text, see [8]
and [6] for more information.