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 $h^3$.

$$y(t+h) = y(t) + \frac{h \cdot (y^{(1)}(t) + y^{(1)}(t+h))}{2}$$ (1.6)

For ordinary differential equations, the trapezoidal rule is an application of the $\theta$ 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