Having a hard time understanding Runge-Kutta integration method

  • Thread starter PCosmin89
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  • #1
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Greetings,

As I am in vacation now there is no way to ask my teacher so I will resort to this forum, I searched and didn't find it on the forum ( hope I haven't skipped anything ) so here goes my question:

For the Runge-Kutta integration methods, I am really puzzled as to what the 'f' function is, for example one of the variables for the xn+1 is f (tn + h⁄2, xn + h⁄2 b), what exactly does the 'f' signify ?

Thank you in advance,

Parvulescu Cosmin
 

Answers and Replies

  • #2
D H
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The Runge Kutta methods (not method) are various ways to solve the initial value problem

[tex]\begin{aligned}
\frac{dx}{dt} &= f(t, x) \\
x(t_0) &= x_0
\end{aligned}[/tex]

One widely-used name for f is the derivative function.
 
  • #3
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The Runge Kutta methods (not method) are various ways to solve the initial value problem

[tex]\begin{aligned}
\frac{dx}{dt} &= f(t, x) \\
x(t_0) &= x_0
\end{aligned}[/tex]

One widely-used name for f is the derivative function.
Thank you for the explanation, I have it cleared now !

Also, if anyone else reads this and needs help with RK4, I have found http://www.youtube.com/watch?v=hGN54bkE8Ac" crash-course which also contains a followup example ! Good luck
 
Last edited by a moderator:
  • #4
Just only formula in the video?

In RK2 we have geometrical interpretation for K1 and K2.
Do we have similar interpretation for K1, K2, K3, K4 in the 4th order Runge-Kutta methods?
 
  • #5
HallsofIvy
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Runge-Kutta is a "predictor-corrector" method. K1 is the slope of the tangent line at xn. That is used to "predict" the value at xn+1 (as in the basic Euler method) and K2 is the slope there. We then correct by using the average of K1 and K2 as slope to predict the value halfway between xn and xn+1 and find the slope, K3, there. Using that slope we predict a new value halfway between xn and xn+1 and find the new slope, K4, there. Finally, we use the (weighted) average of those four slopes to find the value of the function at xn+1.
 
  • #6
So it looks like the interpretation for the K's in RK4 are similar to that of Modified Euler's Method, a variant of RK2.

But something is not right here. The book that I have Schaum Easy Outline on Differential Equation page 109 state that RK4 is not a predictor-corrector method.

Another book that I read state that we cannot estimate the error when using RK4. Is this true?
 
  • #7
D H
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The standard RK4 method can be viewed as a predictor-corrector method in a sense. The first three steps collectively yield a prediction of the value at the end of the integration step. The final step provides a correction to this predicted value.

It is a stretch, and not a very useful one.
 

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