Previous Transient Solutions (Explicit Euler)

Cheers NGSolve Forum,

I need some help understanding an issue I am encountering with regard to previous solutions inside a bi-linear/linear form. To demonstrate the problem I have coded up two versions of a Finite Element Transient Poisson (a non-dimensional heat equation). The first uses implicit Euler time-stepping, and the second explicit Euler.

The only difference between these two codes is the linear/bilinear forms. The implicit Euler solution converges as expected to a steady-state solution (the details of the model are included in the doc-string), while the explicit Euler does nothing (no change when time-stepping).

The program was designed to show convergence to a steady-state solution using ever-decreasing time-steps.

This problem identifies my lack of knowledge of Finite Element methods, or implementation in NGSolve, and help with either would be greatly appreciated.

James

Hi James

here are a couple of comments which might help you:

- You can move outside of the time-stepping loop (but inside the TaskManager), as the matrix does not change over time. At the moment this is quite a lot of redundant computational effort.
- Doing is unnecessary (if you zero out the Griduction at initialisation ) as Dirchlet DOFs remain unchanged when solving the system.
- To copy the data from one GridFunction to another GridFunction (of the same space) I recommend doing . As far as I understand, GridFunction.Set is designed to interpolate a CoefficientFunction into the discrete function, see the documentation .

Now to the explicit Euler part. At the moment you are missing the explicit evaluation of the diffusion operator. This could be done for example by doing BUT: Explicit methods are only stable for very small time-steps. So a direct comparison might not be possible, since the explicit Euler might not be stable for you chosen time-steps. For example you might see that the solution looks good for a few time steps and then explodes.

Best wishes,
Henry

Cheers hvwahl,

Thanks so much for your quick reply and advice. I implemented all of it, and you were spot on for all of it. Thanks.

I realize I uploaded the wrong version of the explicit Euler originally. The version below with your advice implemented is correct. I was able to use these to build a Crank-Nicolson version, and then an adaptive time-stepping version (which was my ultimate goal).

James