I have a question about Heun's numerical method. I have the understanding that Heun's method is basically an improved Euler method which uses Euler as a predictor step and find the values of $y$ at the next steps and take average of the two.
However, I was thinking, what if I create two predictor steps?
For instance:
a) One called $y_n^*$ that predicts $y$ at $\frac{h}{10}$
$y_n^*=y_n+\frac{h}{10}f(y_n)$
b) Another called $y_{n+1}^{**}$ that predicts the value at $\frac{2h}{10}$
$y_{n+1}^{**}=y_n+\frac{2h}{10}f(y_n)$
Finally: the actual would be calculated by $y_{n+1}=y_{n}+h[\omega(y_n^*)+(1-\omega)f(y_{n+1}^{**})]$ , $\omega\in\mathbb{R}$
For instance, we can make $\omega = \text{"anything to make truncation error at least one and particular omega"}$ ($\omega=\frac{1}{2}$) will give us truncation error two. But I am not exactly sure I understand this completely.
How is it (if it is) better than having one predictor step?
In case anything isn't clear:
- $h$ is the step size
- and the interval is [$x_n$, $x_{n+1}$]
Thank you.
$\newcommand{\D}{\mathit{\Delta}}$ You are actually following along the lines that Karl Heun in 1900 was exploring. In the second order case, he considered to construct the next value $y+\D y$ as $$ \D y=\sum \alpha_mf(x+ϵ_m\D x,\,y+ϵ_m\D'_m y)\D x \\ \text{ where }\D'_m y=f(x,y)\D x $$ By comparing Taylor expansions he found the order conditions $$ \sum α_m=1,~~\sum α_mϵ_m=\frac12 $$ and that it is not possible to reach all third order terms in the Taylor expansion of $y(x+\D x)-y(x)$.
So it is perfectly valid to take 3 terms and $ϵ_m=0,\frac1{10},\frac2{10}$. The second order condition $α_2+2α_3=5$ and $α_1=1-α_2-α_3$ can then be satisfied in multiple ways.
For third order methods Heun expanded this scheme to $$ \D y=\sum \alpha_mf(x+ϵ_m\D x,\,y+ϵ_m\D'_m y)\D x \\ \text{ where }\D'_m y=f(x+ϵ'_m,y+ϵ'_m\D''_m y)\D x,\\ \D''_m y=f(x,y)\D x. $$ Now the Taylor expansions match up to third order if $$ \sum α_m=1,~~\sum α_mϵ_m=\frac12,~~\sum α_mϵ_m^2=\frac13,~~\sum α_mϵ_mϵ'_m=\frac16 $$ The first method for this order uses two terms and $ϵ_1=0$ and results in Heuns 3rd order method $$ \D y=\frac14\Bigl(f(x,y)+3f(x+\tfrac23\D x, y+\tfrac23\D'y)\Bigr)\D x\\ \text{ where }\D'y=f(x+\tfrac13\D x, y+\tfrac13f(x,y)\D x)\D x $$
With 3 terms one gets the original 3rd order method of Carl Runge (1895) as one among a large variety of configurations $$ \D y=\frac16\Bigl(f(x,y)+4f(x+\tfrac12\D x, y+\tfrac12f(x,y)\D x)+f(x+\D x, y+\D 'y)\Bigr)\D x\\ \text{ where }\D 'y=f(x+\D x, y+f(x,y)\D x)\D x $$
This rather expansive and repetitive notation was brought into a more compact framework by M. Wilhelm Kutta in 1901