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Runge-Kutta formula/derivation

The Runge-Kutta method has the desirable property of not requiring actual evaluation of the derivatives. The precise form of the function is often not known, or may be very costly to evaluate. A Runge-Kutta method is formed by approximating the second order Taylor polynomial

( is the step size) by

This is done in [2][pages 242-243], and the highlights follow below.

1. To match the coefficients, and , Taylor expand : where the remainder term is

   

2. Match the coefficients of similar terms between and

   This results in the Midpoint method given by

   

   Note, the remainder term is , making the Midpoint method a second order method. This means that the function being integrated is being approximated by a parabola at each step.

3. The above formula is valid for a first order differential equation. The set of equations given in becomes

   

4. To determine the error associated with this approximation, a higher order approximation is needed. The fourth order Runge-Kutta is given by

   

   With two approximations, the quality of the approximation [2][page 271] is given by

   

   This quality factor can be used to increase or decrease the value of the step size. is the tolerance desired. The new step size is given by