IVP for Parabolic-Elliptic 1D Queries

[prao]

Hi Carlos(or anyone else),

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 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?

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!

Best,
Pooja Rao










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