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I want to derive the 4th order Runge-Kutta method. I want to know how to get the coefficient in very detail.

I've read all book about it, possible duplicate, many journals, sites but all of them not telling anything about choosing this coefficient. How do I get $1, \frac12, \frac16, \frac18$, etc? How do I get the constant in the quadrature and sub-quadrature equations?

Rodrigo de Azevedo
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user516076
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  • Yes, yes, yes, yes, and yes. I see that twice. But that's not answering my question. Anyway thank you for the link. Btw, my main question is how do we get some $a_1+a_2+a_3+a_4=1$??? I'm tired looking for this answer. If the answer is comparing the taylor series, yes, i've read it on Wikipedia. But, how do i get $a_1$? It just comes out miraculously. Like appearing miraculously in trees form. – user516076 Nov 17 '19 at 12:27
  • I've seen that one too... well, anyway many thanks for your times. I think i have another question. The last one maybe, if it's clear, i won't ask again. Just forget about the constants. I've read on Wikipedia, RK-1 has 1 coefficient, RK-2 has 2, RK-3 has 3, RK-4 has 4, RK-5 has 6, RK-6 has 7, RK-7 has 9, RK-8 has 11 coefficient. why? Why RK-5 has 6 instead of 5, RK-6 has 7 instead of 6, RK-8 has 11 instead of 8. How do i generalized it on high order Runge Kutta. I don't see the pattern. – user516076 Nov 17 '19 at 13:05
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    What you are asking have pretty obvious answers if you worked them out. Try starting with RK-1, why do we make certain choices? To make it first order? Why 1? Questions like that, and do it for RK-2, RK-3. If you have no idea, try substituting in the known constants and see what happens. Once you actually do all that, you will see how and why you get the constants and why the terms you need does not increase linearly. – David Nov 17 '19 at 13:20
  • @David It's pretty obvious for you but not to me. :'( i just taught the formula by my teacher, i wasn't taught how to derive it. My teacher just gives me the formula and now my job is derive the fourth RK. I don't need the whole answer, i just want to know how to get the RK-coefficients. Thanks. – user516076 Nov 17 '19 at 13:43
  • https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/coefficients-for-the-study-of-rungekutta-integration-processes/457E8B0413C29B9D7BBCD7D2A45A23D5 links to a PDF file with a very relevant paper from the 1960's. Found by typing "Finding runge kutta coefficients" in Bing. – gnasher729 Nov 17 '19 at 19:35
  • Possibly a duplicate question of yours, https://math.stackexchange.com/q/3436705/115115, and a related one for references answering that exact same question, https://math.stackexchange.com/q/3435352/115115. The history is not as organic as one would like to reconstruct, it goes straight from the Euler method as integration approximation to the systematic method construction via Taylor expansions of even the second order methods, see https://math.stackexchange.com/a/2527316/115115 (with the possible exception of the original 3rd order method by Runge). – Lutz Lehmann Nov 18 '19 at 11:37
  • In view of your previous questions, what exactly do you want to know? What does your teacher mean or what do you understand your teacher to have demanded with "derive the fourth RK"? What is the exact wording of that task? Is it some short homework or some large study project? Is reconstructing the derivation of the Fehlberg, Cash-Karp, Dormand-Prince methods also inside the scope of this project? // The relation of stages and order is still an open question, what is known is that the number of independent order conditions grows faster than the number of parameters in an explicit method. – Lutz Lehmann Nov 18 '19 at 12:05
  • Just a short homework actually. Doesn't really affect on my grades. As i said in the previous question, there are many way to choose the parameter as initial value. For example, what i've learned from Heun Method (RK-2), we miraculously choose $\frac 1 2$, and choosing $c_2=c_3=\frac{1}{2}$ in RK-4. And there is no explain about that. How to choose the parameter? Well, we are facilitated by some journals and books, so we definitely know what the coefficient is. But what if i want to generalize this? How do i know how much conditions in 15 th order, 19th order, 100th order? – user516076 Nov 18 '19 at 12:51
  • Again, some books, and journals just skip the explanation about the number of the conditions, the value of the coeffocient selection, they directly gave the final result. – user516076 Nov 18 '19 at 12:55

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I did not say it was obvious to me. I was saying it would be obvious after you worked out the derivation of the lower RK's. Start with the Euler method, and justify why you would want to choose coefficients in a certain way. As the previous commenter mentioned, and provided papers for, it often has to do with the order of approximation the method requires, in this case, 4.

David
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