Fourth Order Runge-Kutta

The fourth-order Runge-Kutta method (RK4) is the most widely used Runge-Kutta method, achieving fourth-order accuracy with four gradient evaluations per step.

Stages

$$k_1=g(t_i,y_i)$$ $$k_2=g\left(t_i+\frac{dt}{2},y_i+\frac{dt}{2}{k}_1\right)$$ $$k_3=g\left(t_i+\frac{dt}{2},y_i+\frac{dt}{2}{k}_2\right)$$ $$k_4=g(t_i+dt,y_i+dt\cdot k_3)$$

Update

$$y_{i+1}=y_i+\frac{dt}{6}(k_1+2k_2+2k_3+k_4)$$

The weights $(1,2,2,1)$ give a weighted average of gradients at the start, two midpoints, and the end of the step. The midpoint evaluations are double-weighted.

Order

Fourth-order: Global truncation error $O(d{t}^4)$, Local truncation error $O(d{t}^5)$.

RK4 is conditionally stable (see Stability of a method), but the stability region is much larger than for lower-order explicit methods, making it practical for most non-stiff problems.

Runge-Kutta methods | Midpoint method | Ralston method | Order of a method | Stability of a method | Global truncation error