Stability Experiments for Backward Euler¶
In [1]:
import numpy as np
import matplotlib.pyplot as pt
We'll integrate
$$ y'=\alpha y$$
with $y'(0) = 1$,
using Backward Euler.
Here are a few parameter settings that exhibit different situations that can occur:
In [23]:
#alpha = -1; h = 0.1; final_t = 20
#alpha = -1; h = 1; final_t = 20
alpha = -1; h = 1.5; final_t = 20
#alpha = 1; h = 0.1; final_t = 20
#alpha = 1; h = 2; final_t = 20
We specify the right-hand side and the initial condition:
In [24]:
t_values = [0]
y_values = [1]
def f(y):
return alpha * y
Integrate in time using Forward Euler:
In [25]:
while t_values[-1] < final_t:
t_values.append(t_values[-1] + h)
y_values.append(y_values[-1]/(1-h*alpha))
And plot:
In [26]:
mesh = np.linspace(0, final_t, 100)
pt.plot(t_values, y_values)
pt.plot(mesh, np.exp(alpha*mesh), label="true")
pt.legend()
Out[26]:
