Runge-Kutta Methods
Runge-Kutta methods evaluate the slope at several trial points inside each step and combine them in a weighted average matched to the Taylor expansion; classical RK4 achieves fourth-order accuracy (error ~ h^4) with four function evaluations per step. Adaptive variants pair two orders (e.g. Dormand-Prince in solve_ivp's RK45) to estimate local error and adjust step size automatically.
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Runge-Kutta methods evaluate the slope at several trial points inside each step and combine them in a weighted average matched to the Taylor expansion; classical RK4 achieves fourth-order accuracy (error ~ h^4) with four function evaluations per step. Adaptive variants pair two orders (e.g. Dormand-Prince in solve_ivp's RK45) to estimate local error and adjust step size automatically.
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