Absolute stability question -

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SUMMARY

This discussion centers on the concept of absolute stability in the context of Numerical Solutions of Ordinary Differential Equations (ODEs) using MATLAB. The user presents a graph comparing the exact solution and the approximate solution with N = 70, indicating that stability is achieved at this point. Key observations include a decreasing saw-tooth pattern with smaller errors as N increases, particularly noting that errors decrease for values above N = 64, leading to convergence. The discussion highlights the importance of clarifying whether stability refers to critical points of differential equations or the stability of numerical methods.

PREREQUISITES
  • Understanding of Ordinary Differential Equations (ODEs)
  • Familiarity with MATLAB for graphing and numerical analysis
  • Knowledge of numerical stability concepts in time-integration schemes
  • Experience with error analysis in numerical methods
NEXT STEPS
  • Research the definition of absolute stability in numerical methods for ODEs
  • Explore MATLAB functions for solving ODEs, such as ode45
  • Learn about the stability criteria for different numerical integration methods
  • Investigate error analysis techniques specific to numerical solutions of ODEs
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Students and professionals in applied mathematics, particularly those studying numerical methods for ODEs, MATLAB users, and anyone interested in understanding the stability of numerical integration techniques.

elle
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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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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:

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