Issue on Runge Kutta Fehlberg algorithm

Question

I have wrote a code for Runge-Kutta 4th order, which works perfectly fine for a system of differential equations:

import numpy as np
import matplotlib.pyplot as plt
import numba
import time
start_time = time.clock()
 

@numba.jit()
def V(u,t):
    x1,dx1, x2, dx2=u
    ddx1=-w**2 * x1 -b * dx1
    ddx2=-(w+0.5)**2 * x2 -(b+0.1) * dx2
    return np.array([dx1,ddx1,dx2,ddx2])


@numba.jit()
def rk4(f, u0, t0, tf , n):
    t = np.linspace(t0, tf, n+1)
    u = np.array((n+1)*[u0])
    h = t[1]-t[0]
    for i in range(n):
        k1 = h * f(u[i], t[i])    
        k2 = h * f(u[i] + 0.5 * k1, t[i] + 0.5*h)
        k3 = h * f(u[i] + 0.5 * k2, t[i] + 0.5*h)
        k4 = h * f(u[i] + k3, t[i] + h)
        u[i+1] = u[i] + (k1 + 2*(k2 + k3) + k4) / 6
    return u, t

u, t  = rk4(V,np.array([0,0.2,0,0.3]) ,0,100, 20000)

print("Execution time:",time.clock() - start_time, "seconds")
x1,dx1,x2,dx2 = u.T
plt.plot(x1,x2)
plt.xlabel('X1')
plt.ylabel('X2')
plt.show()

The above code, returns the desired result:

And thanks to Numba JIT, this code works really fast. However, this method doesn't use adaptive step size and hence, it is not very suitable for a system of stiff differential equations. Runge Kutta Fehlberg method, solves this problem by using a straight forward algorithm. Based on the algorithm (https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta%E2%80%93Fehlberg_method) I wrote this code which works for only one differential equation :

import numpy as np

def rkf( f, a, b, x0, tol, hmax, hmin ):

    a2  =   2.500000000000000e-01  #  1/4
    a3  =   3.750000000000000e-01  #  3/8
    a4  =   9.230769230769231e-01  #  12/13
    a5  =   1.000000000000000e+00  #  1
    a6  =   5.000000000000000e-01  #  1/2

    b21 =   2.500000000000000e-01  #  1/4
    b31 =   9.375000000000000e-02  #  3/32
    b32 =   2.812500000000000e-01  #  9/32
    b41 =   8.793809740555303e-01  #  1932/2197
    b42 =  -3.277196176604461e+00  # -7200/2197
    b43 =   3.320892125625853e+00  #  7296/2197
    b51 =   2.032407407407407e+00  #  439/216
    b52 =  -8.000000000000000e+00  # -8
    b53 =   7.173489278752436e+00  #  3680/513
    b54 =  -2.058966861598441e-01  # -845/4104
    b61 =  -2.962962962962963e-01  # -8/27
    b62 =   2.000000000000000e+00  #  2
    b63 =  -1.381676413255361e+00  # -3544/2565
    b64 =   4.529727095516569e-01  #  1859/4104
    b65 =  -2.750000000000000e-01  # -11/40

    r1  =   2.777777777777778e-03  #  1/360
    r3  =  -2.994152046783626e-02  # -128/4275
    r4  =  -2.919989367357789e-02  # -2197/75240
    r5  =   2.000000000000000e-02  #  1/50
    r6  =   3.636363636363636e-02  #  2/55

    c1  =   1.157407407407407e-01  #  25/216
    c3  =   5.489278752436647e-01  #  1408/2565
    c4  =   5.353313840155945e-01  #  2197/4104
    c5  =  -2.000000000000000e-01  # -1/5

    t = a
    x = np.array(x0)
    h = hmax

    T = np.array( [t] )
    X = np.array( [x] )
    
    while t < b:

        if t + h > b:
            h = b - t

        k1 = h * f( x, t )
        k2 = h * f( x + b21 * k1, t + a2 * h )
        k3 = h * f( x + b31 * k1 + b32 * k2, t + a3 * h )
        k4 = h * f( x + b41 * k1 + b42 * k2 + b43 * k3, t + a4 * h )
        k5 = h * f( x + b51 * k1 + b52 * k2 + b53 * k3 + b54 * k4, t + a5 * h )
        k6 = h * f( x + b61 * k1 + b62 * k2 + b63 * k3 + b64 * k4 + b65 * k5, \
                    t + a6 * h )

        r = abs( r1 * k1 + r3 * k3 + r4 * k4 + r5 * k5 + r6 * k6 ) / h
        if len( np.shape( r ) ) > 0:
            r = max( r )
        if r <= tol:
            t = t + h
            x = x + c1 * k1 + c3 * k3 + c4 * k4 + c5 * k5
            T = np.append( T, t )
            X = np.append( X, [x], 0 )

        h = h * min( max( 0.84 * ( tol / r )**0.25, 0.1 ), 4.0 )

        if h > hmax:
            h = hmax
        elif h < hmin:
            raise RuntimeError("Error: Could not converge to the required tolerance %e with minimum stepsize  %e." % (tol,hmin))
            break

    return ( T, X )

but I'm struggling to convert it to a function like the first code, where I can input a system of differential equations. The most confusing part for me, is how can I vectorize everything in the second code without messing things up. In other words, I cannot reproduce the first result using the RKF algorithm. Can anyone point me in the right direction?

Answer 1

I'm not really sure where your problem lies. Setting the not given parameters to w=1; b=0.1 and calling, without changing anything

T, X = rkf( f=V, a=0, b=100, x0=[0,0.2,0,0.3], tol=1e-6, hmax=1e1, hmin=1e-16 )

gives the phase plot

The step sizes grow as the system slows down as

which is the expected behavior for an unfiltered step size controller.