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In article <1993Jan4.141352.5937@fys.ruu.nl>, walsteyn@fys.ruu.nl (Fred Walsteijn) writes:
|> I have just read that the NAG routine D02BAF is based on
|> a Runge-Kutta-Merson embedded formula.
|> Is it the same formula as mentioned in the books of
|> Lambert (1991) and Butcher (1987) ?
|> 
|> If it is, does that mean that its error truncation estimate
|> is ``easily defeated'' (cite, as I remember it from one of
|> the abovementioned books) ?
|> The reason is that the ``high order'' component has a high
|> (fifth) order of accuracy only for linear problems.
|> It cannot have fifth order of accuracy for all (e.g. nonlinear) problems
|> as it has only five stages (Butcher, 1987).
|> 
|> Thanks for any clarifying remarks...
|> -----------------------------------------------------------------------------
|> Fred Walsteijn                                | Internet: walsteyn@fys.ruu.nl
|> Institute for Marine and Atmospheric Research | FAX:      31-30-543163
|> Utrecht University, The Netherlands           | Phone:    31-30-533169


Most error estimates for integrators are based on what is called
the "linear test equation". It is extremely difficult if not
impossible to obtain local truncation error estimates for 
nonlinear problems. Normally error estimates for nonlinear
problems are obtained by linearizing the equation(s).

As a general rule, there will be a loss of stability and
accuracy when integrators are used for nonlinear equation(s).
This is true of the Runge-Kutta Merson also.  Conversely
a higher order accurate method for a linear problem will be
accurate for a nonlinear problem.

From an implementation point of view, the Runge-Kutta 
family of integrators are robust, accurate and stable. I
have worked with the four stage Runge-Kutta, the Runge-Kutta
Merson and the six stage Runge-Kutta Fehlberg and have found
them to quite reliable. Again, if there are exponentially 
growing terms, then an error analysis breaks down. 

If someone out there has any references on non-linear
stability theory and error estimates for non-linear 
problems, please post them.

Hope this helps somewhat.

Regards


Makarand
makarand@puma.larc.nasa.gov

In article <1993Jan4.141352.5937@fys.ruu.nl> walsteyn@fys.ruu.nl (Fred Walsteijn) writes:
>I have just read that the NAG routine D02BAF is based on
>a Runge-Kutta-Merson embedded formula.
>Is it the same formula as mentioned in the books of
>Lambert (1991) and Butcher (1987) ?
>
>If it is, does that mean that its error truncation estimate
>is ``easily defeated'' (cite, as I remember it from one of
>the abovementioned books) ?
>The reason is that the ``high order'' component has a high
>(fifth) order of accuracy only for linear problems.
>It cannot have fifth order of accuracy for all (e.g. nonlinear) problems
>as it has only five stages (Butcher, 1987).
>

I believe the formulas are the same, though it would be easy enough
to check the NAG documentation to find out.  Shampine showed that 
Merson's pair for general non-linear problems can be viewed as a (3,4) 
pair (i.e. embedded formulas of orders 3 and 4) with a certain "safety" 
factor multiplying the error estimate.  It has been used extensively for 
non-linear problems and generally performs well.  
-- 
Regards,
Mike Hosea (mhosea@seas.smu.edu)

In article <1ia4m9INN2h2@rave.larc.nasa.gov> makarand@gazelle.larc.nasa.gov (Makarand A. Kulkarni) writes:
>Most error estimates for integrators are based on what is called
>the "linear test equation". It is extremely difficult if not
>impossible to obtain local truncation error estimates for 
>nonlinear problems. Normally error estimates for nonlinear
>problems are obtained by linearizing the equation(s).

If I understand you correctly, this isn't true.  Error estimators 
for integrators are usually based on subtracting a higher order 
result from a lower order result to get an asymptotically correct
estimate of the error in the lower order result.  The technique is
valid for linear and nonlinear problems.   

You must be thinking about linear vs. general (nonlinear) stability
analysis, since the "linear test equation" is used in that context,
and general (nonlinear) stability analysis is a lot harder than 
the linear stability analysis.
-- 
Regards,
Mike Hosea (mhosea@seas.smu.edu)

In article <1993Jan5.173151.21651@seas.smu.edu>, mhosea@seas.smu.edu (Mike Hosea) writes:
|> 
|> In article <1ia4m9INN2h2@rave.larc.nasa.gov> makarand@gazelle.larc.nasa.gov (Makarand A. Kulkarni) writes:
|> >Most error estimates for integrators are based on what is called
|> >the "linear test equation". It is extremely difficult if not
|> >impossible to obtain local truncation error estimates for 
|> >nonlinear problems. Normally error estimates for nonlinear
|> >problems are obtained by linearizing the equation(s).
|> 
|> If I understand you correctly, this isn't true.  Error estimators 
|> for integrators are usually based on subtracting a higher order 
|> result from a lower order result to get an asymptotically correct
|> estimate of the error in the lower order result.  The technique is
|> valid for linear and nonlinear problems.   
|> 
|> You must be thinking about linear vs. general (nonlinear) stability
|> analysis, since the "linear test equation" is used in that context,
|> and general (nonlinear) stability analysis is a lot harder than 
|> the linear stability analysis.
|> -- 
|> Regards,
|> Mike Hosea (mhosea@seas.smu.edu)


Yep, you are right. I was talking about stability analysis.
Your point is well taken.

Regards
Makarand
email: makarand@puma.larc.nasa.gov.