Runge Kutta finite difference of differential equations

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Discussion Overview

The discussion centers on the application and derivation of the total variation diminishing (TVD) Runge-Kutta methods for solving differential equations. Participants explore the mathematical formulation, coefficients, and optimality conditions associated with these methods, as well as references to relevant literature.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asks how to derive the TVD Runge-Kutta method and the coefficients such as 3/4 and 1/4 in the equations presented.
  • Another participant notes that Runge-Kutta refers to a family of methods rather than a single method and provides a reference to an article explaining TVD Runge-Kutta.
  • There is a question about the establishment of variables ##\alpha_2## and ##\alpha_3## in the context of the coefficients used in the methods, with a suggestion to consult a specific numerical analysis textbook for details.
  • A participant mentions that the derivation of the condition for TVD, specifically c = min##\left( \frac {\alpha_{ik}} {\beta_{ik}} \right)##, is presented as a lemma in a referenced article, but the proof is not provided in the discussion.

Areas of Agreement / Disagreement

Participants express differing levels of access to resources and literature, leading to some uncertainty about specific derivations and proofs. There is no consensus on the derivation of the coefficients or the optimality conditions, and multiple viewpoints on the understanding of the TVD Runge-Kutta methods are present.

Contextual Notes

Some participants reference specific sections of textbooks and articles for further clarification, indicating that the derivations and proofs may depend on the definitions and assumptions made in those texts. The discussion includes unresolved questions about the mathematical details and conditions for TVD methods.

mertcan
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Hi PF, initially I would like you to focus on that link https://books.google.com.tr/books?id=Dkp6CwAAQBAJ&pg=PA389&lpg=PA389&dq=runge+kutta+method++is+tvd+proof&source=bl&ots=47ULQDVwcC&sig=e2zjdnXENJ7WxBbrf6hXkSouvLI&hl=tr&sa=X&ved=0ahUKEwjU5Z2XsbXZAhUMCMAKHWpnATQ4ChDoAQhKMAQ#v=onepage&q=runge kutta method is tvd proof&f=false starting from page 385 there is a equation: $$\mu_t = L(\mu) = -f(\mu)_x$$ and apply second or third order OPTIMAL TVD(total variation diminishing) Runge Kutta method to above equation. For instance towards third order OPTIMAL TVD(total variation diminishing) Runge Kutta method it is written that $$\mu^(1) = \mu^n + \Delta_t L(\mu^n)$$ $$\mu^(2) = 3/4\mu^n + 1/4\mu^(1) + 1/4\Delta_t L(\mu^(1))$$ $$\mu^(n+1) = 1/3\mu^n + 2/3mu^(2) + 2/3\Delta_t L(\mu^(2))$$ And this TVD Runge Kutta method is totally different from that form Runge Kutta for instance in http://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node5.html

So my question is how do we derive the TVD Runge Kutta method?? How do we derive the coefficients such as 3/4 or 1/4 in the equation ##\mu^(2) = 3/4\mu^n + 1/4mu^(1) + 1/4\Delta_t L(\mu^(1))## ALSO there is optimal situation for that form of Runge Kutta, How do we derive the optimality in that form of TVD Runge Kutta??
 
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Runge-Kutta is not a method, but a family of methods. The article that explains TDV Runge-Kutta is 10.1090/S0025-5718-98-00913-2

By the way, the link you gave to Google Books is not accessible to me.
 
DrClaude said:
Runge-Kutta is not a method, but a family of methods. The article that explains TDV Runge-Kutta is 10.1090/S0025-5718-98-00913-2

By the way, the link you gave to Google Books is not accessible to me.
@DrClaude I looked into your link and I have already that pdf file which I downloaded from that link http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00913-2/S0025-5718-98-00913-2.pdf. But I have a QUESTION here : On page 81 at the bottom part, there are NEW terms that are not introduced to us on previous pages. For instance it says $$c_{10} = \alpha_2$$ or
$$c_{21} = \frac {\alpha_3 * \left( \alpha_3-\alpha_2 \right)} { \alpha_2*\left( 2-3 * \alpha_2 \right)} $$
Could you tell me HOW the variables called ##\alpha_3## and ##\alpha_2## are established?? Where do they come from?? What is the aim of writing in terms of just 2 different variables??

And still that link https://books.google.com.tr/books?id=Dkp6CwAAQBAJ&pg=PA389&lpg=PA389&dq=runge+kutta+method++is+tvd+proof&source=bl&ots=47ULQDVwcC&sig=e2zjdnXENJ7WxBbrf6hXkSouvLI&hl=tr&sa=X&ved=0ahUKEwjU5Z2XsbXZAhUMCMAKHWpnATQ4ChDoAQhKMAQ#v=onepage&q=runge kutta method is tvd proof&f=false is unaccessible to you @DrClaude ??
 
Last edited:
mertcan said:
Could you tell me HOW the variables called ##\alpha_3## and ##\alpha_2## are established?? Where do they come from?? What is the aim of writing in terms of just 2 different variables??
You will have to consult Ralston, A First Course in Numerical Analysis, section 5.6-3. The details are given there. Coming up with these coefficients is not trivial. It is shown in that book that 3rd-order RK methods can be reduced to a two-parameter family (the ##\alpha_2## and ##\alpha_3##).
 
DrClaude said:
You will have to consult Ralston, A First Course in Numerical Analysis, section 5.6-3. The details are given there. Coming up with these coefficients is not trivial. It is shown in that book that 3rd-order RK methods can be reduced to a two-parameter family (the ##\alpha_2## and ##\alpha_3##).
@DrClaude unfortunately, I do not have the relevant book, it seems you have the book, so if you don't mind could you just share the relevant part with me?? or I can share my mail with you and you send me via mail also
 
Hi, again I have a question about runge kutta finite difference but now it is about the TVD(total variation diminishing) case of runge kutta. In this link http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.145.6413&rep=rep1&type=pdf on page 74 (lemma 1.1) it says Runge Kutta method ( which is ##\mu^i = \sum_{k=0}^{i-1}(\alpha_{ik}*\mu^k + \Delta_t*\beta_{ik}*L(\mu^k)))## is TVD under CFL coefficient c = min##\left( \frac {\alpha_{ik}} {\beta_{ik}} \right)## Besides, I am aware of the fact that in order to know whether or not a scheme is TVD then conditions related to Harten theorem is used, but in Runge Kutta scheme Harten theorem conditions are NOT used to make scheme be TVD. So HOW do we arrive at the condition c = min##\left( \frac {\alpha_{ik}} {\beta_{ik}} \right)##? What is the proof if it??
@DrClaude, I am waiting your responses too...
 
Last edited:
mertcan said:
So HOW do we arrive at the condition c = min##\left( \frac {\alpha_{ik}} {\beta_{ik}} \right)##? What is the proof if it??
This is presented as a lemma in the Gottlieb and She article mentioned earlier. They reference She and Osher [12]. You will have to look up that reference (I don't have access to it).
 

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