Implicit Trapezoid Method

For $\dot{y}(t)=g(t,y(t))$, then…

$$y_{i+1}\approx y_i+\frac{\Delta t}{2}\left(g(t_i,y_i)+g(t_{i+1},y_{i+1})\right)$$

It is implicit since $y_{i+1}$ appears on both sides, hence this equation must be rearranged to isolate $y_{i+1}$ before implementation, analogously to the implicit Euler method. It is 2nd-order, with global truncation error $O(\Delta t^2)$.

Proof: Trapezoid Is Symmetric

The key observation is that $y_i$ and $y_{i+1}$ play symmetric roles in the trapezoid equation. Rearranging:

$$y_i\approx y_{i+1}-\frac{\Delta t}{2}\left(g(t_i,y_i)+g(t_{i+1},y_{i+1})\right)$$

This is exactly the original equation with $i\leftrightarrow i+1$ and $\Delta t\to -\Delta t$, confirming symmetry.