Runge Kutta finite difference of differential equations

In summary, the Runge Kutta finite difference method is a numerical method used for solving differential equations. It is a higher-order method that provides a more accurate approximation of the solution compared to other numerical methods such as Euler's method. It can be used for both ordinary and partial differential equations, and is particularly useful for solving stiff equations. However, its accuracy depends on the order of the method and the chosen step size, and in some cases, other numerical methods may be more accurate. While it can be more computationally expensive, its accuracy makes it a preferred method for certain types of differential equations.
  • #1
mertcan
345
6
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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  • #2
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.
 
  • #3
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 ??
 
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  • #4
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##).
 
  • #5
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
 
  • #6
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...
 
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  • #7
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).
 

Related to Runge Kutta finite difference of differential equations

1. What is the Runge Kutta finite difference method?

The Runge Kutta finite difference method is a numerical method used for solving differential equations. It is an iterative process that involves approximating the solution of a differential equation by calculating a sequence of small steps.

2. How does the Runge Kutta method differ from other numerical methods?

The Runge Kutta method is a higher-order method, meaning it provides a more accurate approximation of the solution compared to other numerical methods such as Euler's method. It also allows for larger step sizes, making it more efficient.

3. Can the Runge Kutta method be used for any type of differential equation?

Yes, the Runge Kutta method can be used for both ordinary and partial differential equations. It is particularly useful for solving stiff equations, which are equations that have solutions that change rapidly over a small interval.

4. Is the Runge Kutta method always more accurate than other numerical methods?

No, the accuracy of the Runge Kutta method depends on the order of the method and the step size chosen. In some cases, other numerical methods may provide a more accurate solution.

5. Is the Runge Kutta method computationally expensive?

Compared to other numerical methods, the Runge Kutta method can be more computationally expensive due to its higher-order nature. However, the tradeoff is a more accurate solution, making it a preferred method for certain types of differential equations.

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