Ralston Method

The Ralston method is a second-order Runge-Kutta method designed to minimise the bound on the Local truncation error.

$$k_1=g(t_i,y_i)$$ $$k_2=g\left(t_i+\frac{2}{3}dt,y_i+\frac{2}{3}dt\cdot k_1\right)$$ $$y_{i+1}=y_i+dt\left(\frac{k_1}{4}+\frac{3k_2}{4}\right)$$

Order

Second-order: Global truncation error $O(d{t}^2)$, Local truncation error $O(d{t}^3)$.

Like the Midpoint method, the Ralston method uses two evaluations of $g$ per step. The difference lies in where the intermediate point is sampled ($2dt/3$ rather than $dt/2$) and in the weighted combination of $k_1$ and $k_2$. The Ralston method is preferred when minimising the local error bound is important.

Runge-Kutta methods | Midpoint method | Fourth order Runge-Kutta | Local truncation error | Order of a method