Solution to ordinary differential equation

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Discussion Overview

The discussion revolves around the choice of the 4th order Runge-Kutta method for solving ordinary differential equations compared to lower-order methods. It explores aspects of accuracy, computational cost, and the prevalence of different methods in literature.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that the 4th order Runge-Kutta method is preferred due to its greater accuracy compared to 2nd and 3rd order methods.
  • Others argue that while 2nd and 3rd order methods are quicker, they are less exact, making the 4th order more cost-effective in terms of accuracy.
  • There is a mention of the computational limitations of higher-order methods, which may take too long to compute.
  • One participant raises a question about the lack of literature on the 3rd order Runge-Kutta method, speculating on its possible absence due to the accuracy increase associated with higher orders.
  • Another participant reflects on the nature of Runge-Kutta methods, noting that they approximate changes through clever weighting of function values.

Areas of Agreement / Disagreement

Participants generally agree on the accuracy advantages of the 4th order method over lower orders, but there is uncertainty regarding the prevalence and utility of the 3rd order method, indicating a lack of consensus on its significance in literature.

Contextual Notes

Limitations include the dependence on computational resources and the potential impact of finite arithmetic on the accuracy of results, which remains unresolved.

monty37
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why is the 4th order Runge -Kutta method widely used than the 2nd or 3rd,for
solving ordinary differential equations?
 
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Cost-effectiveness.

Although 2. and 3.order Runge-Kutta are quicker than 4th order, they are much less exact.

For orders higher than 4, those take too long time to compute.

On another note:

Although I won't vouch for at which order this will become significant, the upper limit of an approximate scheme in terms of exactness will be when the finite arithmetic of the computer starts messing with the answers we want.
 
so even if the involved differential equations contains different complex functions,
due to greater accuracy ,the 4th order(RK) method is chosen.
 
arildno said:
Cost-effectiveness.

Although 2. and 3.order Runge-Kutta are quicker than 4th order, they are much less exact.

For orders higher than 4, those take too long time to compute.

On another note:

Although I won't vouch for at which order this will become significant, the upper limit of an approximate scheme in terms of exactness will be when the finite arithmetic of the computer starts messing with the answers we want.

I agree with you. Just that I never see RK3 formula in the literatures :wink:. Why is that so?
 
matematikawan said:
I agree with you. Just that I never see RK3 formula in the literatures :wink:. Why is that so?

Probably because R-K increases by two steps of accuracy each time.
It's been a long time since I had a glancing look at the procedure, and have forgotten if 3.order r-k is even possible. :smile:
What I remember is that r-k approximates changes due to the values of the derivatives by clever weighting of function values, using a nesting principle.
 
Last edited:

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