Having a hard time understanding Runge-Kutta integration method

[PCosmin89]

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 x_n+1 is f (tn + h⁄2, xn + h⁄2 b), what exactly does the 'f' signify ?

Thank you in advance,

Parvulescu Cosmin

 

[D H]

The Runge Kutta methods (not method) are various ways to solve the initial value problem

$$\begin{aligned}<br /> \frac{dx}{dt} &= f(t, x) \\<br /> x(t_0) &= x_0<br /> \end{aligned}$$

One widely-used name for f is the derivative function.

 

[PCosmin89]

The Runge Kutta methods (not method) are various ways to solve the initial value problem

$$\begin{aligned}<br /> \frac{dx}{dt} &= f(t, x) \\<br /> x(t_0) &= x_0<br /> \end{aligned}$$

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

 

[matematikawan]

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?

 

[HallsofIvy]

Runge-Kutta is a "predictor-corrector" method. K1 is the slope of the tangent line at x_n. That is used to "predict" the value at x_n+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 x_n and x_n+1 and find the slope, K3, there. Using that slope we predict a new value halfway between x_n and x_n+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 x_n+1.

 

[matematikawan]

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?

 

[D H]

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.