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#### The Trapezoidal Rule

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.
| Explicit | Implicit
First Order | Forward-Euler, exponential Euler | Backward-Euler
Second Order | 2nd-Order Runge-Kutta | Trapezoidal rule
Higher Order | Adams-Bashforth, Runge-Kutta-Fehlberg | Adams-Moulton, Gear

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