Re: IVP for Parabolic-Elliptic 1D Queries

[c.]

On 5 Mar 2014, at 12:22, prao <address@hidden> wrote:

> Hi Carlos(or anyone else),
Hi, sorry for the late reply!

> I gave a quick read to couple of papers including the one listed in the
> reference for matlab pdepe function
> (http://epubs.siam.org/doi/pdf/10.1137/0911001) and the one listed on the
> project ideas (http://www.pdecomp.net/Scholarpedia/MOLfinal.pdf).

I have no idea who posted this latter reference on the wiki, it is not 
particulrly
interesting nor well written, and is too generic to be informative about the 
specific
problem at hand.

> I have a
> few preliminary questions/thoughts about the problem. Could you please
> correct me or throw some more light or both? My thoughts are a little
> unorganized and I am going to ramble on for a bit, so please bear with me! 
> 
> 1. What method are we trying to use for semi-discretization of spatial
> variable? Finite differencing or finite volume or does it  not matter much
> because we are in 1D? 
> 2. How much influence should boundary conditions have on  choosing  a method
> in #1? 
> 3. What's the plan for stiff ODEs?
> 3. What order of accuracy are we looking for?
> 4. What are we going to do about the singularities that comes from 1/x term
> for the cylindrical and spherical coordinates?
> 5. How different the approach should be for parabolic and elliptic?
> 6. What method do we want to solve the algebraic system of equations in?
> 7. What language do we want to implement it in?

These are all interesting quetsions, and they have a very broad scope.
They give me the impression, though, that your are approaching this project
in a way that is of very little interest for the Octave community.

It seems you are probably not aware that there are already quite a few options 
in Octave and Octave Forge packages for solving IBVP for PDEs in any 
dimensions, 
and the FEnics interface offers access to an even broader range of options.

Therefore we (or at least I) have no interest at all in this project if
what you want to do is just choose an algorithm that you like and implement it.

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
bvp4c
bvp5c


> The matlab paper uses the Galerkin and Petrov-Galerkin method while the
> other resource uses Runge-Kutta type methods. 
> I think we could use RK as we are not solving hyperbolic equations, so we
> don't need any fancy dynamic grid. Also Runge-Kutta (explicit) would be
> easier to code and is memory efficient. Although memory efficiency may not
> be our biggest concern as we are in 1D.
> Both  finite element and finite difference are local methods so I am not
> sure if they deal with the boundary conditions very differently.
> 
> It seems like Galerkin deals with the singularity better by placing the mesh
> point on them, this is not possible for finite differencing where the grid
> is fixed. For RK, we could use adaptive step size (Runge-Kutta-Fehlberg
> method) to resolve this problem.
> 
> For stiff ODEs, there's Galerkin-Petrov method with second order accuracy
> and then there are implicit RK( and also stabilized RK). 
> 
> There are special RK methods out there for parabolic PDEs called parabolic
> RK, etc. Maybe use a specialized elliptic method. Or do we just want to
> decide on one method and then use it across the board.
> 
> Use Newton's method for non-linear algebraic equations. 
> 
> I think C++ or C would probably be better(faster) to implement it in.
> 
> Could you also suggest some other relevant papers/books? Thanks in advance!


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
2) Finding out if there is any pat of those algorithms already implemented in 
Octave/Octave Forge [1]
3) Finding out what remains to be done to complete the implementation of the 
algorithms and to give them a compatible interface.


> Best,
> Pooja Rao

HTH,
c.

[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 ...