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$$x_n = x_0 + h\sum_{k=0}^{n-1}f(t_0 + \alpha_kh, x_k)$$ where $$x_k = x_0 + h\sum_{k=0}^{k-1} \beta_{kj} f(t_0 + \alpha h, x_j)$$

I am given equations to find $a$, $b$, and $c$, but my understanding (probably wrong) was $x_n$ (each step size iterated through the range, $[a,b]$, $nh=b-a$) is found using the 4 K formulas where it uses $h$ and $1/2$ for the classical example. He wants different $\alpha$, $\beta$, and $c$ values, and gives us the formulas to find them.

My question is, how do these values match? I will attach images. Basically, I don't understand what the $kj$ subscript of beta is. Also, how is $x_j$ found?

Original image I got formula from

Lutz Lehmann
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Kenny Coding
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  • Please type the question in latex/mathjax rather than posting an image.... – Aatmaj Apr 10 '21 at 04:00
  • The problem is that what you wrote is wrong, the index of the second sum should be $j$, not $k$. Now the point is that $\beta_{k,j}$ depends on both $k$ and $j$. – Ian Apr 10 '21 at 04:30
  • I will attach an image of where I got the formula from. Its not from our textbook. The problem is he gave us this image and I can't find it anywhere. I book just says its hard to derive these formulas x_x – Kenny Coding Apr 10 '21 at 05:47
  • See https://math.stackexchange.com/questions/3436705/how-to-derive-4th-runge-kutta and the links in my answer there, and also the slides https://www.math.auckland.ac.nz/~butcher/ODE-book-2008/Tutorials/low-order-RK.pdf – Lutz Lehmann Apr 10 '21 at 08:39

1 Answers1

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This scheme is strictly for one step of step size $h$ of an $n$-stage general explicit Runge-Kutta method. The scheme approximates some exact solution $x(t)$ by repeating this one step scheme to get values $x^j\approx x(t^j)$ at regularly spaced points $t^j=t^0+jh$. This is some outer time loop.

What the given equations describe is the internal mechanism of one step of the one-step or Runge-Kutta method. Entering this procedure, it starts with assigning $(t_0,x_0)=(t^j,x^j)$ and returns $x^{j+1}=x_n$ for the time $t^{j+1}$.

The short notation for the mass of coefficients is the Butcher tableau $$ \begin{array}{c|ccc} \alpha_0&\beta_{00}&&0\\ \vdots&\vdots&\ddots\\ \alpha_{n-1}&\beta_{n-1,0}&\cdots&\beta_{n-1,n-1}\\ \hline &\gamma_0&\cdots&\gamma_{n-1} \end{array} \quad\to\left\{\begin{aligned} F_k&=f(t_0+\alpha_kh,x_k)\\ x_k&=x_0+h\sum_{j=0}^{k-1}\beta_{kj}F_j\\ x_n&=x_0+h\sum_{j=0}^{n-1}\gamma_jF_j \end{aligned}\right. $$ In its interpretation to the right one can focus on the "$K$ values", here the $F_k$, or on the intermediate points $x_k$ like on the cited slides.

See Why use Classic fourth-order Runge-Kutta over the 3/8-rule? for some non-standard 4th order schemes by W(ilhelm) Martin Kutta.

Lutz Lehmann
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