Solution to ordinary differential equation

[monty37]

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

 

[arildno]

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.

 

[monty37]

so even if the involved differential equations contains different complex functions,
due to greater accuracy ,the 4th order(RK) method is chosen.

 

[matematikawan]

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 . Why is that so?

 

[arildno]

I agree with you. Just that I never see RK3 formula in the literatures . 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.
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.