How to integrate over the entire bounds of a solution to a PDE at a given time step using PDEPE

I have a diffusion problem, solving for concentration over a spherical volume given diffusion and source terms. I'm interested in calculating the total concentraion over the entire geometry at each time step, which will then be fed into the diffusion parameter, which will then influence the next iteration.

How can I integrate over the entire geometry to find the total concentration at each time step? Do I have use PDEPE for small steps, output the solution, integrate, and restart PDEPE (similar to ODE event functions).

Answer 1

[Torsten]

How can I integrate over the entire geometry to find the total concentration at each time step? Do I have use PDEPE for small steps, output the solution, integrate, and restart PDEPE (similar to ODE event functions).

Yes.

Or use

https://de.mathworks.com/matlabcentral/fileexchange/97437-pde1dm

As far as I remember, if you define a coupling between the PDE and an artificial ODE in this code for all grid points, you get the values of the PDE in all grid points for each time step. This offers the possibility to get the total concentration.

Or assume total concentration over time, integrate, evaluate results, integrate again with the results for total concentration from the first integration, evaluate results,... and so on until convergence.

But if you have a simple 1d pde, why not discretizing in space and use ode15s to solve the resulting system of ODEs in the grid points (method-of-lines) ? Here, you have maximum flexibility.

[Joshua Ferrigno]

Thanks Torsten, pde1dm seems to give the necessary control over the variables I need to perform this integration.

While on a similar topic, I would like to ask: is it possible to access the solution differential at the boundary, inside of the PDE function? Just for a simple example, say that in a similar situation as I mentioned above, the boundary condition on the left was the derivative equal to zero, while the right BC was that the concentration being diffused was equal to zero. If the diffusion parameter was a function of differential of the concentration at the right boundary condition, this cannot be solved alone with pdepe correct?

[Torsten]

If the diffusion parameter was a function of differential of the concentration at the right boundary condition, this cannot be solved alone with pdepe correct?

Correct, this information is not available inside the PDE function of "pdepe". But how can the diffusion coefficient depend on the concentration gradient at another position in space ? Or do you want to deduce the total concentration by the fluxes over the boundaries ?

[Joshua Ferrigno]

I mispoke, another PDE I am solving simultanouesly is dependent on the concentration gradient at the boundary of the diffusion PDE, not the diffusion PDE itself.

Thanks!