ODE Stability A Literature Review

What is Stability?

• Stability means that errors at any stage of the computation are not amplified but are attenuated as the computation progresses.

• In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close to it when x = 0 remains close to it for succeeding values of x.

• If the difference between the solutions approaches zero as x increases, the solution is called asymptotically stable. If a solution does not have either of these properties, it is called unstable.

• We say that a computation is numerically unstable if the uncertainty of the input values is grossly magnified by the numerical method.

When does a ODE becomes unstable?

ODE becomes unstable for larger time steps in explicit method.

Implicit method in unconditionally stable.

PYTHON CODE FOR EXPLICIT METHOD TO OBSERVE ODE STABILITY

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, color=\'green\')
plt.show()

Output of the above code for n=300 (lesser timesteps)

Clearly, the system is stable for a lesser number of time steps.

Output of the above code for n=50 (increased timesteps)

Clearly, system is unstable for more number of time steps.

How does stability affect engineering simulations?

• Stability of ODE is important in Engineering problems because if slight deviations from the mathematical model caused by unavoidable measurement errors do not have any slight effect on the solution, the mathematical equations describing the problem will not accurately predict the future outcome or the solution.

• Some numerical methods for finding roots of equations or solutions for systems of linear equations may diverge rather than converge on the correct answer for certain problems.

• Why would we tolerate this possibility when confronted with design or planning problems? The answer is that these methods may be highly efficient when they work. Thus, trade-off emerges. We must decide if our problem requirements justify the effort needed to apply a method that may not always converge.

Can stability condition be derived for all types of problems?

In general, the stability depends greatly on the form of the function and may be uncontrollable. An example for the conditions of some systems is mentioned below: -

• In Physical stability which includes Linear system and Autonomous system (NOTE: Here A is a matrix):

1. Every solution is stable if all the eigenvalues of A has negative real part.
2. Every solution is unstable if at least one eigenvalue of A has positive real part.

• In Numerical stability, the approximations generated by the method are themselves close to the true solution. The concept of closeness in this context is referred to as numerical stability. In general, to obtain numerical stability, you will need to carefully choose your step size h in the numerical solvers.

• So Yes, stability conditions can be derived for different kinds of problems, but it greatly depends on the nature of the given problem function.