Re: IVP for Parabolic-Elliptic 1D Queries

[prao]

c.-2 wrote
>>> Hi Carlo(or anyone else),
>> Hi, sorry for the late reply!
>> Personally I think this project makes sense only if what you want to do
>> is
>> to implement Matlab compatible solvers.
> 
>> In particular there has been demand on this list for the following
>> solvers
>> that are currently not implemented in Octave:
> 
>> pdepe

I am interested in implementing matlab compatible pdepe.


c.-2 wrote
>> According to what I wrote above, rather than discussing all the possible
>> options
>> for solving 1d differential equations, if you really want to make a
>> successful
>> application on this subject, you should focus on:
> 
>> 1) Finding out what are the algorithms that Matlab solvers are based on

The matlab pdepe page refers to the paper [2], which is the method use to do
spatial discretization during the method of lines formulation.  It is done
using piecewise non-linear Galerkin (regular case) and Petrov-galerkin
(singular case). 
In addition, the time integration is done via ODE15s, which is based on
Numerical Difference Formulas (NDF), developed from BDF.


c.-2 wrote
>> 2) Finding out if there is any part of those algorithms already
>> implemented in Octave/Octave Forge [1]

ODE15s is on the project ideas page to make it matlab compatible, but there
is a simple version [3] which calls DASSL (BDF) and LSODE. There is also
DASPK (adaptive BDF), as you pointed out, which can be used/adapted for our
purposes.  This may require some legwork, I need to become more familiar
with its capabilities before I can make a concrete claim on that.  Both
adaptive BDF (DASPK) and NDF (Matlab) are developed from BDF and I am not
sure how much of difference there is in terms of performance, but I am
looking into that now.

In addition, I am going to look into the tools available in the finite
element libraries to see if there is any function that could aid in the
spatial discretization via Galerkin type methods.


c.-2 wrote
>> 3) Finding out what remains to be done to complete the implementation of
>> the algorithms and to give them a compatible >  interface.

So in summary I think the project consists of the following main stages:

1) Parse Input to determine appropriate method (new code to determine
Galerkin vs Petro-Galerkin)
2) Implement Galerkin algorithms for discretization (through FE library/new
code using [2])
3) Call ODE solver (I'm thinking DASPK at this point, code to call this with
correct options for specific problem)
4) Clean up and produce output

Any comments/suggestions are much appreciated. Thank you Carlo for your
helpful comments!

Best,
Pooja

[1] For example there is a very rough attempt at a draft implementation of
bvp4c in OF, and the
    adaptive BDF algorith used in pdepe for time stepping is the same used
by daspk ...

[2] http://epubs.siam.org/doi/abs/10.1137/0911001

[3] http://savannah.gnu.org/patch/?func=detailitem&item_id=8102#options



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