Self-Implemented Variable-Step Runge-Kutta Method for Solving Strongly Nonlinear Differential Equations

The self-implemented variable-step Runge-Kutta method effectively solves strongly nonlinear differential equations through adaptive step-size control based on local error estimation. This algorithm dynamically adjusts the integration step size according to specified error tolerances, significantly improving numerical solution accuracy. Implementations typically involve embedded Runge-Kutta pairs (like Fehlberg or Dormand-Prince methods) that provide error estimates by comparing results from different-order approximations. Key functions include step-size controllers that use proportional-integral algorithms to maintain stability while minimizing function evaluations. Through optimization of the step-size adjustment strategy and error control mechanisms, the computational efficiency and numerical stability can be further enhanced. This method finds widespread application in numerical computations and maintains significant research value for future developments in stiff differential equation solvers.