Runge-Kutta 2nd and 4th order comparison

runge_kutta.py

'''
Numerical methods for ordinary differential equations are
methods used to find numerical approximations to the solutions
of ordinary differential equations (ODEs). Their use is also 
known as "numerical integration", although this term is sometimes 
taken to mean the computation of integrals.

The Runge–Kutta methods are a family of implicit and explicit 
iterative methods used in temporal discretization for the approximate
solutions of ordinary differential equations.

In this example we are comparing the Runge-Kutta methods 2nd and 4th
orders with the analytical solution. The purpose is to compare the 
efficience of the methods in terms of approximation and, of course,
conclude that the 4th order is much more efficient.

More details: http://mathworld.wolfram.com/Runge-KuttaMethod.html
'''

def runge_kutta2(f, x, y, h):
    k1 = f(x, y)
    k2 = f(x + h, y + h * k1)
    return y + h/2 * (k1 + k2)

def runge_kutta4(f, x, y, h):
    k1 = f(x, y)
    k2 = f(x + h / 2, y + h/2 * k1)
    k3 = f(x + h / 2, y + h/2 * k2)
    k4 = f(x + h, y + h * k3)
    return y + h/6 * (k1 + 2 * k2 + 2 * k3 + k4)

if __name__ == "__main__":
    import math
    '''
        IVP
        y' = y,
        y(0) = 1
    '''
    
    def f(x, y):
        return y
    
    y0 = 1
    h = 0.04
    xn = 0.04 

    yn_1 = runge_kutta2(f, xn, y0, h)
    error = math.exp(xn) - yn_1
    print("Runge-Kutta 2nd order: %s" % yn_1)
    print("Runge-Kutta 2nd order error: %s" % error)

    yn_1 = runge_kutta4(f, xn, y0, h)
    error = math.exp(xn) - yn_1
    print("Runge-Kutta 4th order: %s" % yn_1)
    print("Runge-Kutta 4th order error: %s" % error)

    '''
    output:
        Runge-Kutta 2nd order: 1.0408
        Runge-Kutta 2nd order error: 1.077419238826316e-05
        Runge-Kutta 4th order: 1.0408107733333334
        Runge-Kutta 4th order error: 8.590548272735532e-10
    '''