DerivationODEsadvanced

Derivation: fourth-order Runge-Kutta

Applying Simpson's rule to the integral form of the IVP and approximating the unknown slopes with chained internal evaluations produces the classical RK4 and explains its 1, 2, 2, 1 weights.

Simpson on the integral form

The four RK4 stages replace Simpson's exact slopes by chained internal evaluations.

First integrate the differential equation over the subinterval $[t_k,t_{k+1}]$ :
$\int_{t_k}^{t_{k+1}} y'(\tau)\,d\tau=\int_{t_k}^{t_{k+1}} f\bigl(\tau,y(\tau)\bigr)\,d\tau$
Applying the Fundamental Theorem of Calculus to the left-hand side gives the exact integral form of the IVP:
$y(t_{k+1})=y(t_k)+\int_{t_k}^{t_{k+1}} f\bigl(\tau,y(\tau)\bigr)\,d\tau$
Approximate the integral with Simpson's rule, which uses the endpoints and the midpoint $t_{k+\frac12}=t_k+\frac{h}{2}$ :
$\int_{t_k}^{t_{k+1}} f\,d\tau\approx\frac{h}{6}\Bigl(f(t_k,y_k)+4f\bigl(t_{k+\frac12},y_{k+\frac12}\bigr)+f(t_{k+1},y_{k+1})\Bigr)$
Problem: we know neither $y_{k+\frac12}$ nor $y_{k+1}$ . Runge-Kutta's solution is to estimate them with chained internal slopes, each built by advancing with the previous one:
$\begin{aligned} k_1&=f(t_k,\,y_k)\\ k_2&=f\bigl(t_k+\tfrac{h}{2},\,y_k+\tfrac{h}{2}k_1\bigr)\\ k_3&=f\bigl(t_k+\tfrac{h}{2},\,y_k+\tfrac{h}{2}k_2\bigr)\\ k_4&=f(t_k+h,\,y_k+h\,k_3) \end{aligned}$
Simpson's central slope is approximated by averaging the two midpoint estimates, and the final one by $k_4$ :
$f\bigl(t_{k+\frac12},y_{k+\frac12}\bigr)\approx\frac{k_2+k_3}{2},\qquad f(t_{k+1},y_{k+1})\approx k_4$
Substituting into Simpson, the midpoint weight $4$ splits as $4\cdot\frac{k_2+k_3}{2}=2k_2+2k_3$ , and the classical RK4 appears with its $1,2,2,1$ weights:
$y_{k+1}=y_k+\frac{h}{6}\bigl(k_1+2k_2+2k_3+k_4\bigr)$
A Taylor analysis analogous to Heun's (but up to fourth order) confirms that these internal approximations preserve the $\mathcal{O}(h^5)$ local error, so the method is fourth order.