A family of explicit and implicit single-step numerical integration schemes for ordinary differential equations that approximate the solution by combining weighted evaluations of the equation's right-hand side within each step.

Principle

Advance from tn to tn+1 by computing intermediate slope estimates (stages) at chosen points inside the step and combine them with weights to produce a higher-order accurate increment; order and stability depend on the stage coefficients and whether the scheme is implicit.

Demonstration

The classical fourth-order explicit scheme computes four stages k1=f(tn,yn), k2=f(tn+Δt/2, yn+Δt k1/2), k3=f(tn+Δt/2, yn+Δt k2/2), k4=f(tn+Δt, yn+Δt k3) and sets yn+1 = yn + (Δt/6)(k1+2k2+2k3+k4), producing O(Δt^4) local truncation error for smooth problems.

Misapplication

Using an explicit high-order Runge–Kutta on a stiff system with large eigenvalues of the linearization can require impractically small step sizes and cause instability; also, neglecting adaptive step control on irregular right-hand sides leads to error growth.

Consequence

When chosen appropriately (order and stability region), Runge–Kutta schemes give controlled local and global error with straightforward implementation and can be embedded in adaptive step-size controllers for efficiency.

Reversal

Inverting the idea yields schemes that rely on multistep history rather than internal stages; multistep methods trade internal evaluations per step for dependence on several past solution values.

Boundary

Applies to initial value problems for ordinary differential equations; not directly suitable for boundary-value problems without reformulation, and implicit variants require nonlinear solves at each step.

Semantic Tension

Competing approaches are linear multistep methods, which reuse past steps to achieve order but have different stability constraints; Runge–Kutta methods emphasize per-step stage evaluations to obtain order and simplicity.

Synthesis

Runge–Kutta methods build higher-order single-step integrators by sampling the differential operator inside each step and combining those samples with weights to approximate the exact flow while balancing accuracy and stability.