1
$\begingroup$

We have been asked to prove the following Runge-Kutta method is of order 3

$y_{i+1} = y_i + \frac{K_1}{4} + \frac{3K_3}{4}$

with

$K_1 = h \times f(x_i,y_i)$

$K_2 = h \times f(x_i + \tfrac{h}{3}, y_i + \tfrac{K1}{3})$

$K_3 = h \times f(x_i + \tfrac{2h}{3}, y_i + \tfrac{2K_2}{3})$

We have ideas from this answer -Determining Runge-Kutta Order - but are struggling to apply our system to this method.

Thanks

$\endgroup$
1
$\begingroup$

According to the man the Butcher tables are named after (see these slides), the conditions for order 3 are \begin{align} b_1+b_2+b_3&=1\\ b_2c_2+b_3c_3&=\frac12\\ b_2c_2^2+b_3c_3^2&=\frac13\\ \text{and}\quad b_3a_{32}c_2=\frac16 \end{align} which gives $$b_2c_2(c_3-c_2)=\frac12c_3-\frac13$$ which for $c_2=\frac13$ and $c_3=\frac23$ results in $b_2=0$ and $b_3=\frac34$ which finally has $b_1=\frac14$, $a_{32}=\frac23$, $a_{31}=0$.


This is a (the) third order Heun method, as the type of method Karl Heun (1900) considered were based on combining slope iterations of the form $Δ^m_\nu y = f(x+ε^m_\nuΔx,y+ε^m_\nuΔ^{m+1}_\nu y)Δx$, $m=0,...,s_\nu$ with $ε^{s_\nu}_\nu=0$ into a final update $Δy=\sum \alpha_\nuΔ^0_\nu y$. This third order method Heun, p. 30 gave as

und hieraus resultiert die für die Anwendungen sehr bequeme Formel $$ IV)\quad\left\{\begin{aligned} Δy &= \frac14\left\{f(x,y)+3f\left(x+\frac23Δx,y+Δ'y\right)\right\}\cdotΔx\\ Δ'y &= \frac23f\left(x+\frac13Δx,y+\frac13f\cdotΔx\right)\cdotΔx \end{aligned}\right. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.