This is the formula of explicit Heun's method $$ y_{i+1}=y_i+hf\left(t_{i+1},y_i+hf(t_i,y_i)+O(h^2)\right)+O(h^3) $$ and I want to prove that this formula is $O(h^3)$.
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$y_{i+1}=y_i+hf\left(t_{i+1},y_i+hf(t_i,y_i)+O(h^2)\right)+O(h^3)=y_i+h(f\left(t_{i+1},y_i+hf(t_i,y_i)\right)+f'(t_{i+1},y_i+hf(t_i,y_i))O(h^2))+O(h^3)=y_i+hf\left(t_{i+1},y_i+hf(t_i,y_i)\right)+O(h^3)$
Pierre Alvarez
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How did you achieve it? – Dante Sep 13 '14 at 12:02
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First order taylor serie approximation? – Pierre Alvarez Sep 13 '14 at 12:16