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Here Heun's method means the explicit endpoint-averaging Runge-Kutta method: take an Euler predictor $Y_2=y_n+hf(t_n,y_n)$, sample $k_2=f(t_n+h,Y_2)$ there, and average it with $k_1=f(t_n,y_n)$ for the output. The second Runge-Kutta stage is evaluated at the predictor, not at the corrected endpoint. In particular it requires no implicit solve as the ODE trapezoidal rule does. Naming conventions vary between books; the formulas specify the method unambiguously.^[ref]

For smooth $f$, a Taylor expansion about $(t_n,y_n)$ gives $k_2=f+h(f_t+f_{y}f)+O(h^2)$, where $f_y$ is the state Jacobian matrix. Consequently the output is $y_n+hf+h^2(f_t+f_{y}f)/2+O(h^3)$, matching the exact solution through degree two by the chain rule. This gives local truncation error $O(h^3)$ and, with the usual bounded smoothness and Lipschitz continuity hypotheses on a fixed finite interval, global error $O(h^2)$. The Euler-Heun embedded pair compares this output with its Euler predictor.

For $y^{\prime }=y$, $y_n=1$ and $h=0.2$, the sampled slopes are $1$ and $1.2$. Transporting the second slope to the starting point would end at $1.24$; the average slope $1.1$ instead ends at $1.22$. The exact exponential function reaches $e^{0.2}\approx 1.22140275816$. The magnified endpoint comparison separates the pair difference $0.02$ from the actual Heun local defect $0.00140275816$.