Fixed-Step Runge-Kutta Method Configuration Issue

I've observed numerous developers seeking guidance on fixed-step Runge-Kutta method configuration, prompting me to offer a more detailed explanation. The Runge-Kutta method is a widely-used numerical approach for solving ordinary differential equations (ODEs). Specifically, the fixed-step method maintains a constant time step throughout the computation while employing Runge-Kutta techniques to solve differential equations. This approach helps reduce cumulative errors and can handle complex differential equation systems effectively. Below, I present a fourth-order fixed-step Runge-Kutta implementation with code annotations for reference.

The following program demonstrates the implementation:

# Fourth-Order Fixed-Step Runge-Kutta Implementation

def rk4(f, x0, t):

# Initialize solution array with initial condition x0 repeated for all time points

n = len(t)

x = np.array([x0] * n)

# Iterate through time steps using fourth-order Runge-Kutta algorithm

for i in range(n - 1):

# Calculate current time step size

h = t[i+1] - t[i]

# Compute four slope estimates (k1-k4) for weighted average

k1 = h * f(x[i], t[i]) # Slope at beginning of interval

k2 = h * f(x[i] + 0.5 * k1, t[i] + 0.5 * h) # Slope at midpoint using k1

k3 = h * f(x[i] + 0.5 * k2, t[i] + 0.5 * h) # Refined slope at midpoint using k2

k4 = h * f(x[i] + k3, t[i] + h) # Slope at end of interval using k3

# Combine slopes using classical weighting (1:2:2:1 ratio)

x[i+1] = x[i] + (k1 + 2 * k2 + 2 * k3 + k4) / 6

return x # Return complete solution trajectory

I hope this implementation proves helpful. Please feel free to contact me with any questions regarding the algorithm or code implementation.