Stability Experiments for Forward Euler¶
In [3]:
import numpy as np
import matplotlib.pyplot as pt
We'll integrate
$$ y'=\alpha y$$
with $y'(0) = 1$,
using forward Euler.
Here are a few parameter settings that exhibit different situations that can occur:
In [22]:
#alpha = 1; h = 0.1; final_t = 20
#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 = 2; final_t = 20
#alpha = -1; h = 2.5; final_t = 20
We specify the right-hand side and the initial condition:
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t_values = [0]
y_values = [1]
def f(y):
return alpha * y
Integrate in time using Forward Euler:
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while t_values[-1] < final_t:
t_values.append(t_values[-1] + h)
y_values.append(y_values[-1] + h*f(y_values[-1]))
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while t < t_final:
y = y + h * y * np.sin(3*t)
t += h
And plot:
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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[25]:
In [27]:
mesh = np.linspace(0, final_t, 15)
pt.plot(mesh, y_values-np.exp(alpha*mesh), label="true")
pt.legend()
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