Fourth-Order Runge-Kutta Method for Solving Systems of Ordinary Differential Equations

In the fields of mathematics and computer science, the fourth-order Runge-Kutta method (commonly known as RK4) is a widely used numerical technique for solving systems of ordinary differential equations (ODEs). This method employs a discrete time-stepping approach, utilizing a sequence of calculated steps to approximate the solution of differential equations. The RK4 algorithm is generally recognized as a high-precision numerical integration technique, offering superior stability and accuracy compared to many other methods.

The implementation involves four key stages per time step: calculating slope estimates at the beginning, midpoint, and end of the interval, followed by a weighted average to determine the next value. A typical code implementation would include:

- Initialization of time steps and initial conditions
- Loop structure for iterative calculation
- Four slope calculations (k1, k2, k3, k4) representing different interval estimates
- Weighted average computation using the formula: y_{n+1} = y_n + (k1 + 2k2 + 2k3 + k4)/6

The relative simplicity of implementation allows for straightforward programming in various computational environments. Consequently, RK4 finds extensive application in scientific computing and engineering calculations, particularly in simulations requiring high numerical precision.