Implicit Trapezoid Method

The implicit trapezoid method is a second-order implicit method that averages the gradients at the start and end of each step.

$$y_{i+1}=y_i+\frac{dt}{2}\left(g(t_i,y_i)+g(t_{i+1},y_{i+1})\right)$$

Like the Implicit Euler method, $y_{i+1}$ appears on both sides, so the equation must be solved implicitly at each step.

Order

Second-order: Global truncation error $O(d{t}^2)$, Local truncation error $O(d{t}^3)$.

Symmetry (Time-Reversibility)

The implicit trapezoid method is symmetric: if we swap indices $i\leftrightarrow i+1$ and negate $dt$, we recover the same equation. This means the method is time-reversible, which is a desirable property for problems where the physics is reversible.

Note: Heun’s method (explicit trapezoid) uses a predicted ${\tilde{y}}_{i+1}$ from an Euler step in place of the true $y_{i+1}$. It is NOT the same as the implicit trapezoid method and is NOT symmetric.

Stability

The implicit trapezoid method is unconditionally stable - there is no restriction on step size. This follows from the implicit treatment of $g(t_{i+1},y_{i+1})$.

Implicit Euler method | Explicit Euler method | Runge-Kutta methods | Stability of a method | Order of a method