Runge kutta method, Mathematics

As noted, Euler's method is little used in practice, as there are much better ways of solving initial value problems. By better, we mean, "able to achieve a result of the same precision using a larger step size". [Euler's method is also unstable for some problems where the step size can take you outside the physical domain of the function and the solution runs away to infinity.]

To improve on Euler's method, we will use the fourth-order Runge-Kutta method. This method requires four evaluations of the differential at each step, but often allows a much larger step size to achieve the same result. The method can be summarised as:

2432_Runge-Kutta method.png

where we have written h in place of the step size Δx.

  • Write a program to use the Runge-Kutta method of Eq. to solve the equation of motion of the projectile given in Eq.
  • Repeat your analysis of the step size required to achieve a given accuracy in the result. Doyou see an improvement? By how much?
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