Add Support for Symmetric Interior Penalty (SIP) Method for Parabolic Terms in Trixi.jl

[kwkwon13]

I am currently working on implementing the advection-diffusion equation in Trixi.jl. Upon reviewing the source code, I have observed that the semidiscretization for parabolic terms is handled within solvers/dgsem_tree/dg_1d_parabolic.jl. Specifically, the function rhs_parabolic! utilizes arguments such as dg::DG for the solver and parabolic_scheme, although it appears that parabolic_scheme doesn't play any role in the current implementation. The diffusion term seems to default to the classical Bassi-Rebay 1 (BR1) formulation.

While the BR1 formulation is currently supported and the LDG formulation remains on the TO-DO list, I would like to implement the Symmetric Interior Penalty (SIP) method for diffusion terms, as described in https://doi.org/10.1007/978-3-642-22980-0.

Proposed Changes:

1. Add support for the Symmetric Interior Penalty (SIP) method within the solvers/dgsem_tree/dg_1d_parabolic.jl module.
2. Update the parabolic_scheme argument to recognize and handle this new formulation, allowing users to select between BR1, LDG, and SIP formulations.

[jlchan]

Hi @kwkwon13; yes, you're correct that the diffusion defaults to BR-1. We'd welcome your help in implementing SIP; it would likely be easiest to do this in the flux formulation.

Just out of curiosity, can I ask why you're interested specifically in SIP? It would be relatively straightforward to add BR-1 with an LDG-like penalty, which behaves similarly to SIP in many circumstances.