Midpoint Method for Ordinary Differential Equations in Python

#----------------------------------------------------------------------
#
# midpointmethod.py
#
# calculate the curve which is the solution to an ordinary differential
# equation with an initial value using the midpoint method
#
# Paul Soper
#
# April 24, 2016
#
#-----------------------------------------------------------------------

import math
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

# we will use the differential equation y'(t) = y(t).  The analytic solution is y = e^t.

def y1(t,y):
    return y

def asol(t):
    return math.exp(t)

yasol = np.vectorize(asol)

h = 0.1
t0 = 0.0
y0 = 1.0

t = np.arange(0.0, 5.0, h)
y = np.zeros(t.size)
y[0] = y0

for i in range(1, t.size):
    y[i] = y[i-1] + h*y1(t[i-1] + h/2, y[i-1] + (h/2.0)*y1(t[i-1],y[i-1]))

    

plt.plot(t,y,'r-',t,yasol(t),'b-')