Runge-Kutta code does not match the built-in method

Question

I am trying to implement the runge-kutta method to solve the Lotka-Volterra system, but the code (below) does not work correctly. I followed the recommendations I found in other StackOverflow themes, but the results do not match the built-in Runge-Kutta method, for example, using the rk4 method available in Pylab. Can anybody help me?

import matplotlib.pyplot as plt
import numpy as np
from pylab import *

def meurk4( f, x0, t ):
    n = len( t )
    x = np.array( [ x0 ] * n )    

    for i in range( n - 1 ):

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

        k1 = h * f( x[i], t[i] )
        k2 = h * f( x[i] + 0.5 * h * k1, t[i] + 0.5 * h )
        k3 = h * f( x[i] + 0.5 * h * k2, t[i] + 0.5 * h )
        k4 = h * f( x[i] + h * k3, t[i] + h)

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

    return x

def model(state,t):

    x,y = state     

    a = 0.8
    b = 0.02
    c = 0.2
    d = 0.004
    k = 600

    return np.array([ x*(a*(1-x*k**-1)-b*y) , -y*(c - d*x) ]) # corresponds to [dx/dt, dy/dt]

# initial conditions for the system
x0 = 500
y0 = 200

# vector of time
t = np.linspace( 0, 50, 100 )

result = meurk4( model, [x0,y0], t )
print result

plt.plot(t,result)

plt.xlabel('Time')
plt.ylabel('Population Size')
plt.legend(('x (prey)','y (predator)'))
plt.title('Lotka-Volterra Model')
plt.show()

---

I just updated the code following the comments. So the function meurk4:

def meurk4( f, x0, t ):
        n = len( t )
        x = np.array( [ x0 ] * n )    

        for i in range( n - 1 ):

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

            k1 = h * f( x[i], t[i] )
            k2 = h * f( x[i] + 0.5 * h * k1, t[i] + 0.5 * h )
            k3 = h * f( x[i] + 0.5 * h * k2, t[i] + 0.5 * h )
            k4 = h * f( x[i] + h * k3, t[i] + h)

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

        return x

Now (fixed):

def meurk4( f, x0, t ):
    n = len( t )
    x = np.array( [ x0 ] * n )    

    for i in range( n - 1 ):

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

        k1 = f( x[i], t[i] )
        k2 = f( x[i] + 0.5 * h * k1, t[i] + 0.5 * h )
        k3 = f( x[i] + 0.5 * h * k2, t[i] + 0.5 * h )
        k4 = f( x[i] + h * k3, t[i] + h)

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

    return x

However, the results are as follows:

enter image description here

While the buitin rk4 method (from Pylab) leads to the following:

enter image description here

So of course, my code is still incorrect, as its results do not match the rk4 built-in method. Can someone please help me?

Answer 1

, ., , Runge-Kutta 4 RK4 Python

k2 = f( x+0.5*h*k1, t+0.5*h )
...
x[i+1]=x[i]+(k1+2*(k2+k3)+k4)*(h/6)

k2 = h*f( x+0.5*k1, t+0.5*h )

.., x[i+1] , .

---

: - , , x. , , ,

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

. ,

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

. , 1, . ,

# initial conditions for the system
x0 = 500.0
y0 = 200.0

.