Runge-Kutta methods improve time stepping by sampling the ODE's slope at several points within each step instead of using only the slope at the beginning.

For example, the classical fourth-order method computes four slope estimates $k_1,k_2,k_3,k_4$ and combines them:

$$y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4).$$

The intermediate slopes estimate how the derivative changes across the step.

For sufficiently smooth problems, the accumulated error of classical RK4 over a fixed interval decreases proportionally to $h^4$ as the step size is refined.

Runge-Kutta methods are one-step methods: the next value is constructed from the current state and derivative evaluations within the current step, without requiring a stored history of earlier states.

Higher order improves accuracy per step but requires more function evaluations, so efficiency depends on the cost of evaluating the differential equation and the requested tolerance.