Summary of Backward Euler vs. Trapezoidal Rule

For

$$\begin{eqnarray*} f(t)&=&m\,a(t) \;=\;m\,\dot{v}(t) \;=\;\dot{p}(t)\\ [5pt] &=& \lim_{T\to0}\frac{p(t)-p(t-T)}{T} \,\;\approx\;\,\frac{p(t)-p(t-T)}{T} \end{eqnarray*}$$

• Backward Euler (BE)
  $$f_n = \frac{1}{T}\,(p_n - p_{n-1})$$
  is ${\cal O}(T)$ (first-order accurate in $T$ )

• Bilinear Transform, or Trapezoid Rule (TR)
  $$f_n = \frac{2}{T}\,(p_n - p_{n-1}) - f_{n-1},$$
  is ${\cal O}(T^2)$ (second-order accurate in $T$ )

• A continuum of transforms
  $$s=\frac{1+\alpha}{T}\,\frac{1-z^{-1}}{1+\alpha z^{-1}}$$
  exists between BE and TR and can be optimized for the application at hand (see Kurt Werner thesis and Germain and Werner DAFx-15 paper for details--Germain thesis coming soon)