Runge-Kutta Method for Solving Systems of Differential Equations

The Runge-Kutta method is a numerical technique for solving systems of differential equations. Compared to MATLAB's built-in ODE solving commands (such as ode45 or ode15s), the Runge-Kutta method provides superior accuracy and minimal error accumulation through its sophisticated iterative approach. A key advantage of the Runge-Kutta method lies in its adaptive step-size control capability, allowing it to dynamically adjust the integration step based on local truncation error estimates. This makes it highly suitable for various differential equation systems with different stiffness characteristics and precision requirements. The method typically implements multiple function evaluations per step (commonly 4 evaluations for the classical 4th-order version) to achieve higher-order accuracy while maintaining numerical stability. The algorithm's robust stability properties and reliable numerical solutions make it widely applicable in scientific computing and engineering practice. Implementation typically involves calculating weighted averages of slope estimates at different points within each time step, ensuring balanced trade-offs between computational efficiency and solution accuracy.