Fixed-Step Fourth-Order Runge-Kutta Method Algorithm for Solving Differential Equations (Systems)

This article introduces a fixed-step fourth-order Runge-Kutta method algorithm for solving differential equations. This algorithm can be implemented on computers and applied to solve numerous real-world problems in fields such as physics, engineering, economics, and more.

The algorithm's source code is presented below, demonstrating the classical fourth-order implementation:

// Runge-Kutta Method Algorithm

def runge_kutta(f, x, y, h):

k1 = h * f(x, y) // Calculate slope at initial point

k2 = h * f(x + h/2, y + k1/2) // Calculate slope at midpoint using k1

k3 = h * f(x + h/2, y + k2/2) // Calculate slope at midpoint using k2

k4 = h * f(x + h, y + k3) // Calculate slope at end point using k3

return y + (k1 + 2*k2 + 2*k3 + k4) / 6 // Weighted average to compute next y-value

The algorithm employs a four-stage slope calculation process that provides fourth-order accuracy, making it particularly suitable for solving both ordinary differential equations and systems of differential equations with consistent step size. Each k-value represents a different slope estimate, and the weighted combination ensures higher precision compared to lower-order methods.

We hope this article proves helpful and enhances your understanding of differential equation solving methodologies.