Having a hard time understanding Runge-Kutta integration method

In summary: The classical RK4 method is better viewed as a member of the set of explicit Runge-Kutta (ERK) methods.In summary, The Runge-Kutta methods (not method) are various ways to solve the initial value problem \begin{aligned} \frac{dx}{dt} &= f(t, x) \\ x(t_0) &= x_0 \end{aligned} One widely-used name for f is the derivative function. The fourth order Runge-Kutta method (RK4) is a predictor-corrector method and a member of the set of explicit Runge-Kutta (ERK) methods. While not explicitly estimating the error, RK4 does provide a correction to the predicted value at the end
  • #1
PCosmin89
2
0
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
 
Physics news on Phys.org
  • #2
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
D H said:
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
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
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.
 

1. What is Runge-Kutta integration method?

Runge-Kutta integration method is a numerical method used to solve differential equations. It is named after its inventors, Carl Runge and Martin Kutta, and is commonly used in scientific and engineering applications.

2. How does Runge-Kutta integration method work?

The method works by breaking down the problem into smaller steps and using approximations to calculate the value of the solution at each step. The calculated values are then used to update the solution until the desired level of accuracy is achieved.

3. What are the advantages of using Runge-Kutta integration method?

One of the main advantages of this method is its accuracy. It is a higher-order method, meaning that it uses more information from previous steps to calculate the next step, resulting in a more accurate solution. It is also a versatile method that can be applied to a wide range of differential equations.

4. What are the limitations of Runge-Kutta integration method?

One limitation of this method is that it can be computationally expensive, especially for higher order equations or when a high level of accuracy is required. It also requires the step size to be small enough for accurate results, which can be time-consuming in certain cases.

5. Are there any alternatives to Runge-Kutta integration method?

Yes, there are other numerical methods for solving differential equations, such as Euler's method and the midpoint method. Each method has its own advantages and limitations, and the choice of method depends on the specific problem being solved and the desired level of accuracy.

Similar threads

  • Differential Equations
Replies
6
Views
2K
  • Differential Equations
Replies
6
Views
3K
  • Differential Equations
Replies
5
Views
1K
  • Differential Equations
Replies
4
Views
6K
  • Differential Equations
Replies
4
Views
2K
Replies
9
Views
2K
Replies
2
Views
3K
  • Programming and Computer Science
Replies
15
Views
2K
Replies
1
Views
9K
  • Differential Equations
Replies
31
Views
6K
Back
Top