Ralston Method

Similar to the Midpoint method, but modifying by evaluating two-thirds through the interval instead of halfway. This is called Ralston’s method:

$$y_{i+1}\approx y_i+\Delta t\cdot \left(\frac{1}{4}g(t_i,y_i)+\frac{3}{4}g\left(t_i+\frac{2}{3}\Delta t,y_i+\frac{2}{3}g(t_i,y_i)\Delta t\right)\right)$$

However, this is a bit easier to implement by breaking it into intermediate variables, namely $k_1=g(t_i,y_i)$ and $k_2=g\left(t_i+\frac{2}{3}\Delta t,y_i+\frac{2}{3}\Delta t\cdot k_1\right)$:

$$y_{i+1}\approx y_i+\Delta t\cdot \left(\frac{1}{4}{k}_1+\frac{3}{4}{k}_2\right)$$