Absolute stability question -

In summary, the conversation is about a question regarding absolute stability in the class Numerical Solutions of ODEs. The person has produced a graph on Matlab showing the exact solution and approximate solution for a given method with N = 70 when it is just stable. They have made observations about the decreasing saw-tooth solution and decreasing errors for N > 64, with the solution eventually converging. They are unsure if they are on the right track and are looking for additional comments. More details are needed about the specific method being used and the definition of stability for a numerical time-integration scheme.
  • #1
elle
91
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Absolute stability question - urgent!

Hi, I wasn't too sure what section to put this in but I'm currently working on an exercise regarding absolute stability in the class Numerical Solutions of ODEs. Here is the graph i have produced on Matlab and I am suppose to comment on it. The graph shows the exact solution (black line) and the approximate solution of the given method with N = 70 when it is just stable.

http://i16.tinypic.com/2e5ky0l.jpg"

So far, I have made the comments that it can be observed that there is decreasing saw-tooth solution with smaller errors. For any values above N = 64, the errors will continue to decrease and the solution eventually converging.

But I don't know if I am on the right track, can anyone help me out? Any additional comments I have left out would be helpful!
 
Last edited by a moderator:
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  • #2
elle said:
Hi, I wasn't too sure what section to put this in but I'm currently working on an exercise regarding absolute stability in the class Numerical Solutions of ODEs. Here is the graph i have produced on Matlab and I am suppose to comment on it. The graph shows the exact solution (black line) and the approximate solution of the given method with N = 70 when it is just stable.

http://i16.tinypic.com/2e5ky0l.jpg"

So far, I have made the comments that it can be observed that there is decreasing saw-tooth solution with smaller errors. For any values above N = 64, the errors will continue to decrease and the solution eventually converging.

But I don't know if I am on the right track, can anyone help me out? Any additional comments I have left out would be helpful!


It would be helpful if you gave a few more details here. First, when you say stability, are you referring to the stability of a critical point of a differential equation, or the stability of a numerical method for solving a differential equation? Second, what method are you using to solve the equation? What is N? Is it the number of time points in the interval? Do you know the definition for stability of a numerical time-integration scheme?
 
Last edited by a moderator:

FAQ: Absolute stability question -

What is the absolute stability question?

The absolute stability question is a concept in mathematics and engineering that deals with the stability of a system. It asks whether a system, such as a control system or a physical system, will remain stable under all possible conditions and inputs.

How is the absolute stability question related to control systems?

The absolute stability question is closely related to control systems, as it helps determine whether a control system will be able to maintain stability and function properly under all conditions. It is an important consideration in designing and analyzing control systems.

What factors are important in determining absolute stability?

There are several factors that are important in determining absolute stability, including the system's transfer function, the location of its poles and zeros, and the system's gain. These factors can be analyzed using mathematical techniques such as the Nyquist stability criterion or the Routh-Hurwitz stability criterion.

Can a system be both stable and unstable at the same time?

No, a system cannot be both stable and unstable at the same time. The absolute stability question is asking whether a system will remain stable under all possible conditions, so if a system is unstable under any condition, it is not considered absolutely stable.

How is the absolute stability question relevant in real-world applications?

The absolute stability question is relevant in many real-world applications, particularly in engineering and physics. It is important in designing and analyzing control systems, predicting the behavior of physical systems, and ensuring the safety and reliability of various technologies.

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