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Time discretization - RUNGE-KUTTA_EXPLICIT method

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Old   April 16, 2023, 19:30
Default Time discretization - RUNGE-KUTTA_EXPLICIT method
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Hello everyone,

There are 3 methods to time discretization (Euler Explicit, Euler Implicit, and Runge-Kutta Explicit). Is it true that for a time dependent problem I should use an explicit method (and a low CFL)?

And what is the Runge-Kutta explicit method that appears? I understand that this is a method from the RK family and 3rd order, because there are 3 coefficients, but I did not understand what these coefficients are and how they are calculated and what is Runge Kutta alpha coefficients? And why usually in the examples they were (2/3, 2/3, 1)? Is there a reference to where I can read about the method? Or what is the difference between Runge-Kutta alpha coefficients and without the alpha?

Thank you
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Old   April 17, 2023, 02:09
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You derive the RK method by constructing a Taylor expansion for the discrete equation and constructing the Butcher tableau. The coefficients are the weight factors, so choosing them differently leads to different behavior and different RK methods.

You usually learn about this in a first course on numerical methods for ODEs, so if you have the opportunity, you should follow such a course. You can also pick up a good book on the topic, like the book from Butcher, 'Numerical Methods for Ordinary Differential Equations'
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Old   April 17, 2023, 06:04
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Thank you bigfootedrockmidget for your response,

I did learn about RK methods, so I will try to refine my question.

Does the "Runge-Kutta Explicit" option in SU2 refer to the RK 4th order? I didn't find a related document on the SU2 website.

So for example, if I set the coefficients as (1/3,2/3,1) do I get the "3/8-rule"? (the "3/8-rule" is mentioned here, or as (1/2,1/2,1) for the ‘classical Runge–Kutta method’ (also appears in the book you mentioned (thanks for recommending it)).

If this is the case, then the three coefficients we choose are c2,c3,c4, if I understand correctly.

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