why does pdepe adopt Petrov-Galerkin?

pdepe is meant to solve parabolic and elliptic PDEs. Petrov-Galerkin seems to be designed to solve convection dominated ones, why would matlab use this algorithm for solving non-hyperbolic ones?

[Torsten]

Address your question to the creators of "pdepe". Most probably, they found in their tests that it worked well for the problem class. Or they wanted to use a non-standard method to have a basis for a new publication.

[feynman feynman]

creators of pdepe are from mathworks, so I asked here :)

[Torsten]

creators of pdepe are from mathworks, so I asked here :)

No. Responsible for the theoretical approach are the authors of the article:

[1] Skeel, R. D. and M. Berzins, "A Method for the Spatial Discretization of Parabolic Equations in One Space Variable," SIAM Journal on Scientific and Statistical Computing, Vol. 11, 1990, pp.1–32.

TMW won't be able to help in this respect.

[feynman feynman]

right, I meant there must be good reason for mathworks to decide to adopt their Petrov-Galerkin?

[Torsten]

The good reason is that it worked well for the problem class. Nobody in practise cares about the method with which a certain aim is reached.

[feynman feynman]

it just makes one wonder why pdepe and solvepde adopt different FEMs (assuming solvepde adopts a more regular FEM)

[Torsten]

Different university chairs propagate different methods - just consider all the methods used in codes for ordinary differential equations (BDF, Runge-Kutta, Extrapolation, Multistep,...).

"pdepe" is a stand-alone program to solve parabolic-elliptic PDEs in one spatial dimension. So why should the method used not differ from the one of the PDE Toolbox (or to whatever "solvepde" belongs) ?

[feynman feynman]

makes sense, thanks! I wonder if there's any tests on pdepe regarding if it's dissipative or non-dissipative for conservative PDEs (though it's not designed for hyperbolic ones)?

[Torsten]

Again: ask Dr. Skeel and Dr. Berzins.

[feynman feynman]

Thanks for the tips

[Torsten]

I doubt that the code works for equations without a smoothing second-order derivative. Otherwise, the second-order term wouldn't be mandatory: pde p- parabolic e- elliptic, not pde p- parabolic e- elliptic -h hyperbolic.

[feynman feynman]

It actually solves some hyperbolic PDEs correctly but am just not sure in which situations it can't solve well

[Torsten]

It cannot solve hyperbolic pdes because the f-term should not equal 0. Further, you have to specify two boundary conditions for each equation, and for equations of hyperbolic type you need only one. So one boundary condition will be wrong or at most artificial.

Why do you want a code force to solve hyperbolic equations if its name already indicates that it is created for the parabolic-elliptic type ?

If you are in need to solve hyperbolic PDEs, use CLAWPACK:

https://www.clawpack.org/

[feynman feynman]

Many thanks for the suggestion of CLAWPACK, which I'll check out. For pdepe, I don't know if periodic boundary conditions are allowed but if so PDEs having only first order spatial derivatives can also be solved. I think other finite element software packages can't solve first order hyperbolic ones well either, except when periodic boundary conditions are used.

[Torsten]

For pdepe, I don't know if periodic boundary conditions are allowed but if so PDEs having only first order spatial derivatives can also be solved.

Periodic boundary conditions are not possible with pdepe.

As said, setting up a problem with only first-order derivatives is technically possible. But you have to assume a second boundary condition that is mathematically incorrect. And usually - because the first derivative is in essence approximated by a central difference quotient - the results won't be stable.

[feynman feynman]

I wonder if clawpack doesn't work in windows?

[Torsten]

Why not ? But it's not commercial - so it works with FORTRAN or python, not MATLAB.

[feynman feynman]

but their Installation Prerequisites says it only works in

• Linux
• Mac OS X

[Torsten]

I didn't know - thanks for the info. Maybe you could try Cygwin.

When I worked with Clawpack, I just used the FORTRAN files together with mingw under Windows. But the software has grown ...