Adaptive Runge-Kutta methods

Adaptive Runge-Kutta methods are advanced algorithms used to solve differential equations by estimating solutions step-by-step. They automatically adjust the size of each step based on the estimated error, taking smaller steps when the solution changes rapidly and larger steps when it's smoother. This approach improves accuracy and efficiency, making it possible to obtain reliable results without unnecessary calculations. These methods are widely used in scientific and engineering simulations where precise, adaptive control over the numerical process is essential.