a Runge-Kutta method
/ Runge-Kutta (ROONG-uh KOO-tah) /
Euler trusts one slope. Improved Euler and the midpoint method sample two slopes and combine them, and they pay off with second-order accuracy. The natural next thought is: why stop at two? Sample the slope at several cleverly chosen points within the step, weight them by a carefully designed recipe, and combine them into one super-accurate effective slope. That is the whole Runge-Kutta idea — a family of single-step methods built by sampling the right-hand side at several internal points and blending the results.
Precisely, a Runge-Kutta method computes several stage slopes k1, k2, k3, ... within one step. The first is the slope at the start; each later one is the slope evaluated at a point built from earlier stages (a fractional step in t and a corresponding combination of the k's in y). The final update is a weighted average: y_(n+1) = y_n + h * (b1 k1 + b2 k2 + ...). The coefficients — where to sample (the nodes), how to build each stage (the internal weights), and the final weights b_i — are not arbitrary; they are chosen so that the weighted average reproduces the true Taylor expansion of the solution up to as high an order as possible. Match more Taylor terms, get a higher-order method.
Runge-Kutta methods are self-starting (they need only the current point, no history), they let you change the step size freely, and they reach high accuracy with a handful of slope evaluations per step. That combination of robustness and accuracy is why they, and the classical fourth-order RK4 above all, are the default workhorse for non-stiff problems. Improved Euler and midpoint are simply the order-2 members of this family; RK4 is the order-4 member that hits the practical sweet spot of accuracy versus cost.
Improved Euler is the simplest Runge-Kutta method (2 stages, order 2): k1 = f(t_n, y_n), k2 = f(t_n + h, y_n + h k1), then y_(n+1) = y_n + (h/2)(k1 + k2). Choosing more stages and different weights climbs to orders 3, 4, 5 and beyond.
Sample several internal slopes, blend them by a designed recipe — a whole family of orders.
More stages does not automatically mean higher order — beyond order 4, an explicit Runge-Kutta method needs MORE stages than its order, which is one reason RK4 (4 stages, order 4) sits at such a sweet efficiency point.