= Euler-Heun embedded pair
{c}
{title2=$d=h(k_2-k_1)/2$}
The Euler-Heun 1(2) pair combines the <explicit Euler method> and <Heun's method>. Here 1(2) names their respective <order of a numerical method>; the accepted endpoint in this article is the second-order one. Its shared <Runge-Kutta stages> and endpoints are
$$
\begin{aligned}
k_1&=f(t_n,y_n),&k_2&=f(t_n+h,y_n+hk_1),\\
Y^{[1]}&=y_n+hk_1,&Y^{[2]}&=y_n+\frac h2(k_1+k_2),\\
d&=Y^{[2]}-Y^{[1]}=\frac h2(k_2-k_1).
\end{aligned}
\qquad
\begin{array}{c|cc}
0&0&0\\
1&1&0\\\hline
&\tfrac12&\tfrac12\\
&1&0
\end{array}
$$
The first output row in this <Butcher tableau> is <Heun's method>; the second is the <explicit Euler method>. Both outputs cost two evaluations of $f$ together, rather than one for <Euler method> plus two for a separately computed <Heun's method>.
For the <initial value problem> $y'=y$, $y(0)=1$, take $h=0.2$. The first slope is $k_1=1$; the stage state is $1+0.2k_1=1.2$, giving $k_2=1.2$. Therefore
$$
Y^{[1]}=1.2,\qquad Y^{[2]}=1.22,\qquad d=0.02,\qquad y(0.2)=e^{0.2}\approx1.22140275816.
$$
The exact <exponential function> gives signed local defects approximately $0.02140275816$ for the <explicit Euler method> and $0.00140275816$ for <Heun's method>. Their difference is exactly $0.02$. Thus the pair difference estimates the larger Euler defect; calling it the actual Heun defect would miss by a factor about fourteen here.
More generally, a step from $y_n$ on $y'=y$ has
$$
\begin{aligned}
\ell_1&=y_n(e^h-1-h)=y_n\left(\frac{h^2}{2}+\frac{h^3}{6}+O(h^4)\right),\\
\ell_2&=y_n(e^h-1-h-h^2/2)=y_n\left(\frac{h^3}{6}+O(h^4)\right),\\
d&=y_n\frac{h^2}{2}.
\end{aligned}
$$
These original <Taylor expansions> separate three quantities: the accepted <Heun's method> has <local truncation error> $O(h^3)$ and <global error> $O(h^2)$ under suitable <numerical stability>, while the tested <local error estimator> is $O(h^2)$. For $E=|d|/s$ with $s$ held approximately fixed, use $h_{\rm new}=\eta h E^{-1/2}$. A cube-root update would model a different quantity, the actual higher-order local defect.
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