Accuracy

Backward-difference approximation:

$$\begin{eqnarray*} \omega_a &=& \omega_d + {\cal O}(\omega_d^2 T) \end{eqnarray*}$$

Trapezoid rule (bilinear transform):

$$\begin{eqnarray*} \omega_a &=& \omega_d + {\cal O}(\omega_d^3 T^{2}) \end{eqnarray*}$$

• Trapezoid rule (bilinear transform) is second-order accurate in $T$.
• Higher order accuracy obtainable using more neighboring grid points.
• How should these extra grid points be brought in?