Formulas for Numerical Methods

In what follows $f_m = f(x_m)$ and $y_m = y(x_m)$.
Examples
$f_0 = f(x_0)$ , $f_1 = f(x_1)$ , $f_2 = f(x_2)$ ...
$y_0 = y(x_0)$ , $y_1 = y(x_1)$ , $y_2 = y(x_2)$ ...

Trapezoidal Rule of Integration

$\displaystyle \int_{x_0}^{x_m} f(x) dx = h \left( \dfrac{f_0}{2} + f_1 + f_2 + f_3 + ... + f_{m-1}+\dfrac{f_m}{2}\right)$
$h = \dfrac{x_m - x_0}{m}$

Simpson Rule of Integration

The number of intervals must be even and is taken to be $2 m$
$\displaystyle \int_{x_0}^{x_{2m}} f(x) dx = \dfrac{h}{3} \left[ f_0 + 4( f_1 + f_3 + f_5 + ...) + 2 ( f_2 + f_4 +f_6 + .... )+ f_{2m} \right]$
$h = \dfrac{x_{2m} - x_0}{2 m}$

Euler Method for Differential Equations

$y' = f(x,y)$
$y_{n+1} = y_n + h f(x_n , y_n) + O(h^3)$

Second Order Runge-Kutta Method for Differential Equations

$y' = f(x,y)$
$y_{n+1} = y_n + \dfrac{1}{2} (k_1 + k_2) + O(h^3)$
where
$k_1 = h f(x_n , y_n)$
$k_2 = h f(x_n+h , y_n+k_1)$

Fourth Order Runge-Kutta Method for Differential Equations

$y' = f(x,y)$
$y_{n+1} = y_n + \dfrac{1}{6} (k_1 + 2 k_2 + 2 k_3 + k_4) + O(h^5)$
where
$k_1 = h f(x_n , y_n)$
$k_2 = h f(x_n+\dfrac{h}{2} , y_n + \dfrac{k_1}{2} )$
$k_3 = h f(x_n+\dfrac{h}{2} , y_n + \dfrac{k_2}{2} )$
$k_4 = h f(x_n+ h , y_n + k_3)$

Second Order Runge-Kutta Method for Systems of Differential Equations

$y' = f(x,y,z)$     ,     $z' = g(x,y,z)$
$y_{n+1} = y_n + \dfrac{1}{2} (k_1 + k_2) + O(h^3)$
$z_{n+1} = z_n + \dfrac{1}{2} (l_1 + l_2) + O(h^3)$
where
$k_1 = h f(x_n , y_n , z_n)$     ,     $l_1 = h g(x_n , y_n , z_n)$
$k_2 = h f(x_n+h , y_n + k_1 , z_n + l_1)$     ,     $l_2 = h g(x_n+h , y_n + k_1 , z_n + l_1)$

Fourth Order Runge-Kutta Method for Systems of Differential Equations

$y' = f(x,y,z)$     ,     $z' = g(x,y,z)$
$y_{n+1} = y_n + \dfrac{1}{6} (k_1 + 2 k_2 + 2 k_3 + k_4) + O(h^5)$
$z_{n+1} = z_n + \dfrac{1}{6} (l_1 + 2 l_2 + 2 l_3 + l_4) + O(h^5)$
where
$k_1 = h f(x_n , y_n , z_n)$     ,     $l_1 = h g(x_n , y_n , z_n)$
$k_2 = h f(x_n+\dfrac{h}{2} , y_n + \dfrac{k_1}{2} , z_n + \dfrac{l_1}{2})$     ,     $l_2 = h g(x_n+\dfrac{h}{2} , y_n + \dfrac{k_1}{2} , z_n + \dfrac{l_1}{2} )$
$k_3 = h f(x_n+\dfrac{h}{2} , y_n + \dfrac{k_2}{2} , z_n + \dfrac{l_2}{2} )$     ,     $l_3 = h g(x_n+\dfrac{h}{2} , y_n + \dfrac{k_2}{2} , z_n + \dfrac{l_2}{2})$
$k_4 = h f(x_n + h , y_n + k_3 , z_n + l_3)$     ,     $l_4 = h g(x_n + h , y_n + k_3 , z_n + l_3)$

Note: The fourth order Runge-Kutta method for systems of differential equations is more stable. I used it in my PhD work on "Evanescent Field Effects on Pulse Propagation in Transversely Inhomogeneous Waveguides" , page 43, and it gave excellent results.