I usually assess the stability of a numerical scheme for ODEs analyzing the so called characteristic roots of the characteristic equation deriving from the method used.

Basically, depending on the values of the roots and their location on a 2 - D plane it is possible to assess if the method is stable, relatively stable and weakly stable. Considers that in this analysis the whole structure of the numerical scheme is involved and you should the be able to investigate the effects of all your terms (from the lowest to highest order).

I suggest you to read an excellent and easy to study book about these topics:

"Numerical Solution of Ordinary Differential Equations" (K.E. Atkinson, W. Han, D.E. Stewart"

I hope that this information will be useful for your work.

Paolo

Thanks

It is quite nice book i think.

Can you please explain with details what is truncation error and propagated error?

In each step of integration, we approximate the solution with different order algorithms.

In all these algorithms, series is truncated based on the order of the algorithms. And this truncation introduces the error called the Temporal truncation error (TTE). So at each time step this TTE is introduced into the solution, right.

Now Where the propagated part of the error coming from?

Please someone explain this in detail.

I know this is written in detail in different books. But it is not explicitly stated.

Thanks

Vivek

Three type of errors are present when applying a numerical integration scheme: truncation error, roundoff error and the propagation error. 

Truncation error: is function of the number of terms retained in the approximation of the solution with respect to the infinite series expansion. It may be reduced by retaining a larger number of series terms and/or by reducing the time step.

Roundoff error: related to the machine being used. It may be reduced using, for example, a double precision approach.

Propagation error: represented by the previous two errors.

So the entity of error propagated is directly related to the accuracy of the computer being used and to the choice of the numerical method through the truncation error.

The truncation error has a double nature: a local nature represented by the error caused by a single iteration and a global nature related to the cumulative error through many iterations.  

It should be noted that the propagation mechanism is independent from the time step since it is implicitly linked to the nature of the numerical scheme adopted. 

Paolo

 Thanks a lot Paolo.

Your explanation was quite helpful. 

In relation to truncation error,if the number of the terms in the infinite series is large, then truncation error reduces. But Is stability of the solution affected by the presence of the higher order terms? How larger the number of higher order terms lead to the oscillatory nature of the solution?

Truncation error is having global nature. If we take an example of Euler's method, its local truncation error is O(h2) and Cumulative effect of these errors (global effect) is of order O(h). So does it mean that Euler's method is having an accuracy of order O(h)? 

Round off or chopped off error is related to the Machin epsilon, right? Is it accumulated at each iteration?

Coming to the propagation of error,I learned from a book that only round off error gets propagated as time increases. Is it true?

Please go through the above queries with more attention and Please explain thoroughly.

Thanks

Vivek

A simple way of thinking here is the following: In many methods, you have a step size h (or an number of terms N kept in an infinity expansion - you may think of N=O(1/h)). Neglecting round off error, you would expect the more accurate results the small is h. But since there are more operations for smaller h, a larger rounding error will result. So, often an optimal h can be determined such that the accuracy is worse for smaller h due to increased roundoff and for larger h because then the basic method is too inaccurate.

Stability is not only/always related to roundoff errors. If your algorithm amplifies small changes in the input to large changes in the output, then the algorithm may be regarded as unstable. This may also occur for instance in linear algebra, see for instance chapter 2.7 of the most valuable book "matrix computation" of Golub and van Loan. Also, most good text books on numerical analysis discuss stability. I suggest that you first have a look there.

Hi Vivek

Some comments on you  resent questions.

Is stability of the solution affected by the presence of the higher order terms?

No, There is no    direct relation between the stability and truncation error. In fact,  even exact schemes can be unstable.   Round off error does not provide instability too (this is an intrinsic feature of discrete systems)

So does it mean that Euler's method is having an accuracy of order O(h)?

Yes, it is, but the accuracy can be achieved when the stability conditions is fulfilled.

Round off or chopped off error is related to the Machin epsilon, right?

Yes,

Is it accumulated at each iteration?

In the stable algorithm it is bounded. This kind of the error is essential mainly for ill posed (badly conditioned) problems .

Coming to the propagation of error,I learned from a book that only round off error gets propagated as time increases. Is it true?

Round off error is additively involved in the solution similar to a noise source . Actually, the error propagation demonstrates the same dynamics as perturbations of the solution. It is quite trivial at simulations of linear systems while in nonlinear systems it can be more complicated, for example, spectrally selective. Usually it can be clarified in framework of the nonlinear stability analysis.  

Hi,

Yes I had some wrong understanding about the different errors. Questions are

quite trivial.  

Anyway thanks a lot for explaining thoroughly.

So prof Volkov error propagation/blow up is because of unstable nature of algorithm. In stable algorithm, errors remain bounded and never magnify.

Basically stability is associated with how the problem is defined (i.e. problem may be inherently unstable), and the algorithm used to solve that problem. Stability of the algorithm used depends on the time step size selected. Above certain time step size, the numerical solution will be unstable because of spurious roots. And hence error will magnify in time.  

Just to extend the Herbert's comment. Now as step size decreases, truncation error  decreases, but as the number of computation steps increases round off error increases. Hence, as the step size decreases, the total error initially decreases, then passes through a minimum and then again increases. So one has to find out the minimum.

Give some  comments.

Thanks

Vivek