Forward Euler - Numerical Analysis

from pylab import *

The following function implements Euler method.

y_n = y_{n-1} + h f(t_{n-1},y_{n-1})

(1)

def euler(f,t0,T,y0,h):
    N = int((T-t0)/h)
    y = zeros(N)
    t = zeros(N)
    y[0] = y0
    t[0] = t0
    for n in range(1,N):
        y[n] = y[n-1] + h*f(t[n-1],y[n-1])
        t[n] = t[n-1] + h
    return t, y

Example 1

Consider the ODE problem

\begin{align} y' &= y, \qquad t \ge 0 \\ y(0) &= 1 \end{align}

(2)

whose exact solution is

y(t) = \exp(t)

(3)

Right hand side function and exact solution

def f(t,y):
    return y

def yexact(t):
    return exp(t)

We run the Euler method for decreasing values of step size.

t0,y0,T = 0.0,1.0,5.0

te = linspace(t0,T,100)
ye = yexact(te)
plot(te,ye,'--',label='Exact')

H = [0.2,0.1,0.05]

for h in H:
    t,y = euler(f,t0,T,y0,h)
    plot(t,y,label="h="+str(h))

legend(loc=2), xlabel('t'), ylabel('y'), grid(True);

Even visually we observe that the numerical solution is converging towards the exact solution as $h \to 0$ .

Example 2

Consider the ODE problem

\begin{align} y' &= -y + 2 \exp(-t) \cos(2t), \qquad t \ge 0 \\ y(0) &= 0 \end{align}

(4)

with exact solution

y(t) = \exp(-t) \sin(2t)

(5)

Right hand side function and exact solution

def f(t,y):
    return -y + 2.0*exp(-t)*cos(2.0*t)

def yexact(t):
    return exp(-t)*sin(2.0*t)

Solve for a given h

t0,y0 = 0.0,0.0
T  = 10.0
h  = 1.0/20.0
t,y = euler(f,t0,T,y0,h)
te = linspace(t0,T,100)
ye = yexact(te)
plot(t,y,te,ye,'--')
legend(('Numerical','Exact'))
xlabel('t'), ylabel('y'), grid(True)
title('Step size = ' + str(h));

Solve for several h

Study the effect of decreasing step size. The error is plotted in log scale.

hh = 0.1/2.0**arange(5)
err= zeros(len(hh))
for (i,h) in enumerate(hh):
    t,y = euler(f,t0,T,y0,h)
    ye = yexact(t)
    err[i] = abs(y-ye).max()
    semilogy(t,abs(y-ye))
    
legend(hh), xlabel('t'), ylabel('log(error)')

for i in range(1,len(hh)):
    print('%e %e %e'%(hh[i],err[i],log(err[i-1]/err[i])/log(2.0)))

# plot error vs h
figure()
loglog(hh,err,'o-',label="Error")
loglog(hh,hh,'--',label="y=h")
legend(), xlabel('h'), ylabel('Error norm');

5.000000e-02 4.107182e-02 1.035228e+00
2.500000e-02 2.026883e-02 1.018886e+00
1.250000e-02 1.007087e-02 1.009075e+00
6.250000e-03 5.019323e-03 1.004623e+00

9.1Forward Euler