Literature review ODE Stability

Objective:

To discuss the stability of an ODE system.

When does a ODE become unstable:

The ODE becomes unstable for larger timesteps in explicit method. Implict method is unconditionally stable.

Consider a python program for explicit method to observe the stability of ODE.

import math
import matplotlib.pyplot as plt

def f(t,y):
	return -1000*y-math.exp(-t)

t_start=0
t_end=0.1
n=300

dt=(t_end-t_start)/(n-1)
t=0;
y_old=0;
y_new=[]
for i in range(0,n):
	y_new.append(y_old+dt*f(t,y_old))
	y_old=y_new[i]
	t=t+dt

plt.plot(y_new)
plt.show()

Results:

Here n=300 for which the time step is small. So the system is stable.

Here n=50 for which time step increased and system becomes unstable.

How does stability affect engineering simulations:

In engineering simulations if the solution becomes unstable , it will diverge from the required value rather than converging . Due to this error the required solution cannot be obtained for the particular engineering simulations.

Can stability condition be derived for all types of problems:

Stability conditions can be derived for all types of problems. There are two common stability conditions they are

a)eigen value stability condition

b)von neuman stability conditions