pde1dm compared to pdepe

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feynman feynman
feynman feynman 2024-2-2
编辑： feynman feynman 2024-3-5
pde1dm is an extended version of pdepe and it can solve equations pdepe can't solve. I wonder about the differences between these two algorithms in the spatial discretization?
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Torsten
Torsten 2024-2-8
编辑：Torsten 2024-2-8
How should the finite element method produce a result converging to a senseful solution if you use e.g. (exaggerated, I admit), three mesh points for a distance of 100 km ? I can imagine that such a problem could produce Inf or an oscillating solution in the center point during time integration.
And there is no result I remember that says "the finite element method is stable per se". This can't be true because there is no such thing as "the" finite element method.

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Bill Greene
Bill Greene 2024-2-2
编辑：Bill Greene 2024-2-2
There are several differences in the pde1dm spatial discretization compared with pdepe.
For the m=0 case (Cartesian), both pde1dm and pdepe use the standard, linear shape functions. But, by default, pde1dm evaluates the residual using a two-point Gaussian integration rule while pdepe evaluates only at the center of the element. Often this doesn't significantly affect the solution but, for some problems, this allows pde1dm to obtain a converged solution when pdepe cannot.
Of course, by default, this makes pde1dm somewhat slower than pdepe but use of the "Vectorized" option with pde1dm dramatically reduces the computational time.
In evaluating the so-called mass terms (those arising from the c-coefficients), pdepe essentially uses what is often referred to as a "lumped" approach that results in a diagonal mass matrix. pde1dm evaluates these terms using exactly the same approach as the f and s terms ("consistent" approach).
For cylindrical (m=1) and spherical (m=2) geometries, pdepe uses special shape functions, as described in the Skeel and Berzins paper referenced in the pdepe documentation. According to the paper, this improves the convergence near the singular point (x=0). pde1dm does not use these special shape functions.
The above describes the current implementation of pde1dm but may change in the future. Also, pde1dm has some undocumented options that cause pde1dm to produce a discretization essentially the same as pdepe for the m=0 case. But these options may change or be eliminated in future versions; hence they are undocumented for now.
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Torsten
Torsten 2024-3-4
ode15i is an implicit ODE integrator and has a substantially different interface compared to the other solvers. It cannot be replaced by another integrator from the ode suite.
feynman feynman
feynman feynman 2024-3-5
编辑：feynman feynman 2024-3-5
I saw there was an if statement about whether to use ode15i and the code requires that this statement be always true. I wonder if one could add functionality so that other ode solvers can be put into the else statement.

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Torsten
Torsten 2024-2-2
移动：Torsten 2024-2-2
I wonder about the differences between these two algorithms in the spatial discretization?
No difference. pde1dm has the option to add ordinary differential equations to the system of partial differential equations and in the boundary condition part, but this doesn't change the numerical kernel.
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feynman feynman
feynman feynman 2024-2-2
Thank you. However when the same equation is solved the two solvers give quantitative slightly different results, that's why I doubted. If no difference, how does pde1dm manage to be more flexible in incorporating ODEs?
Torsten
Torsten 2024-2-2
You should contact the author of the code - Bill Greene.

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