The three$-$stage RK method given by the tableauThe three$-$stage RK method given by the tableau

$0,0,0,0$

$\frac{1}{2},\frac{1}{2},0,0$

$1,0,1,0$

$,\frac{1}{6},\frac{2}{3},\frac{1}{6}$

is based, like the classical RK method, on the Simpson quadrature rule. Implement this

method and add a corresponding column to Table $16\mathrm{.}2.$ What is the observed order of

the method?

Table $16\mathrm{.}2.$ Errors and calculated observed orders $($rates$)$ for the forward Euler, the explicit

midpoint $($RK$2),$ and the classical Runge$-$Kutta $($RK$4)$ methods.

is based, like the classical RK method, on the Simpson quadrature rule. Implement this

method and add a corresponding column to Table $16\mathrm{.}2.$ What is the observed order of

the method?