Spatial discretization in pdepe compared to others

The spatial discretization in pdepe seems to be a unique one. Could someone please explain it briefly so that it is understandable in terms of how it differs from other finite differences? Also on the stability and accuracy.

[Torsten]

Did you read the reference for "pdepe" ?

[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.

?

It should be a usual finite-element discretization in 1d, shouldn't it ?

[feynman feynman]

yes I did but it doesn't seem to be a finite element one?

[Torsten]

The authors claim in the introduction that they "propose a simple piecewise nonlinear Galerkin / Petrov-Galerkin method that is second-order accurate in space". These methods are summarized under Finite Element Methods as far as I know. But if you have doubts, you can either read the relevant chapters in a book on the numerical solution of partial differential equations or contact Technical Support to get further details.

[feynman feynman]

Thanks. It will be good to get answers from this community because I don't really know how to contact the technical support. If it's a Galerkin / Petrov-Galerkin method, I wonder about the stability and what sorts of hyperbolic equations it can't solve.

[Torsten]

Simply speaking, hyperbolic equations need specific discretization schemes for the flux terms so that the solution remains stable. The schemes used for parabolic-elliptic differential equations don't account for this.

Since there is an interaction of spatial discretization and integration in time with ode45, I think not much about the stability of the integration with "pdepe" can be said.

But why do you need these specific information ? Do you want to compare different pde solvers with respect to their capability in solving pdes of different type ?

[feynman feynman]

thanks a lot and this helps. I want to summarize what matlab can do in solving PDEs. The pdepe and solvepde seem to be the two go to methods. Can we conclude that the latter is always better?

[Torsten]

"pdepe" is for one-dimensional, "solvepde" for two- and three-dimensional problems (as far as I can see).

Define "better" !

Broader range of equations that can be solved, more user-friendly, faster, less memory consumption ? I cannot answer this, but "better" is always related to what is required. And since both codes solve problems of different dimension, I don't know exactly how to compare them.

[feynman feynman]

I guess solvepde can also solve 1d problems, then in terms of dimensions of solvepde is superior to pdepe.

By better, I mean if solvepde can solve all hyperbolic equations that pdepe can't?

[Torsten]

I guess solvepde can also solve 1d problems, then in terms of dimensions of solvepde is superior to pdepe.

If this is the case, my guess is that internally, "pdepe" is called.

By better, I mean if solvepde can solve all hyperbolic equations that pdepe can't?

There is no official MATLAB solver that can solve hyperbolic PDEs.

[feynman feynman]

thank you for all the answers

[feynman feynman]

Is the interpolation in pdepe based on linear FEs for m=0?