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Spatial discretization in pdepe compared to others

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feynman feynman 2024-1-30

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评论： feynman feynman 2024-3-4

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.

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Torsten 2024-1-30

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编辑：Torsten 2024-1-30

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 2024-1-30

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

Torsten 2024-1-30

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编辑：Torsten 2024-1-30

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 2024-2-1

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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 2024-2-1

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编辑：Torsten 2024-2-1

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 2024-2-1

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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 2024-2-1

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编辑：Torsten 2024-2-1

"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 2024-2-2