Timeline for Solvers for odd order PDE finite difference discretisation
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 5, 2022 at 20:56 | vote | accept | lightxbulb | ||
| Feb 4, 2022 at 18:06 | comment | added | lightxbulb | I did some more experiments and it is non-singular for upwind schemes however, as one would expect. I need to look further into how this generalizes to $u_{xxx}$ though. | |
| Feb 4, 2022 at 17:53 | comment | added | lightxbulb | I did some experiments, most notably I formed the matrix even for the discretisation of $u_x$ with its Dirichlet conditions as a toy example, and then computed the eigenvalues of $A^TA$. Turns out that the matrix with Dirichlet conditions is singular, which explains why CGNR was not converging. I still have to figure out what this means for the problem. Whether the conditions are insufficiently many, whether they are unachievable or something else. | |
| Feb 4, 2022 at 1:46 | comment | added | lightxbulb | I will try the hermitian CG approach in the following days and see what I get. I have also considered using a spectral approach, but it looks more painful handling arbitrary Dirichlet constraints with it and would be slower with the multiple FFTs required than with CG with a spatial discretisation, especially for large problem sizes. | |
| Feb 4, 2022 at 1:40 | comment | added | lightxbulb | Thank you for the answer. It would be great if this can be solved in an uncomplicated manner. I ran several tests with explicit Euler in the meantime but they always diverged. The thing that worked was scaling back to $|\frac{du}{dx}| =\partial_t u$ and applying the Rouy-Tourin discretisation with explicit Euler. This resulted in a piecewise constant polynomial as expected. The main issue is that $u_{xxx}$ seems to have the same or worse discretisation problems, and I am not aware of a Rouy-Tourin scheme for 3rd derivatives. $u_{xxx}=0$ is preferable over $|u_{xxx}|=u_t, \, T=\infty$ too. | |
| Feb 4, 2022 at 0:54 | history | answered | davidhigh | CC BY-SA 4.0 |