Robin boundary for electrostatic solver

[ancientflow]

Dear WarpX Developers,
I am reaching out to ask for guidance on adding Robin boundary condition support to WarpX's electrostatic solver.
From my current investigation, Robin boundary conditions are only natively supported in AMReX's cell-centered solver, while WarpX uses a nodal solver for its electrostatic solver. I have considered two possible approaches to address this, and would greatly value your perspective.
The first approach is to implement a cell-centered electrostatic solver, then interpolate the potential onto the node. The second approach would be to extend the existing nodal solver to support Robin boundary conditions directly, which may be more difficult.
Given the difference in implementation complexity, I am leaning toward the first solution. I would like to ask whether you think this approach is technically feasible, or if there is a more recommended way to add Robin boundary condition support to WarpX's electrostatic solver.
Thank you very much for your time and guidance.

[ax3l]

hi @ancientflow ,

Thanks for the issue. While I think @jlvay and @WeiqunZhang can give more guidance, can you share a bit more why Robin boundaries are needed in WarpX and what alternatives would exist to achieve he same goal, e.g., with existing options?

Maybe it's my ignorance speaking from reading the wikipedia article and not having used Robin BCs before :D

[ancientflow]

hi @ax3l ,
Thanks for your response.
The motivation comes from the need to model an "infinity" boundary condition (i.e., vanishing potential at infinity) in electrostatic PIC simulations. The paper doi:10.1088/1361-6595/ac56ec, which studies PIC simulations of the magnetic nozzle of electric thrusters, proposes a Robin-type boundary condition to approximate this behavior.
Regarding alternatives with the existing options in WarpX:

• Dirichlet (grounded) boundaries have been used in previous studies to approximate the far-field condition, but this requires a sufficiently large computational domain, which significantly increases the computational cost.

• Neumann boundaries suffer from a similar domain-size requirement, and in our experience also make MLMG convergence considerably harder.

• In addition, with mesh refinement, the coarser levels may fail to correctly capture the non-trivial potential profile (e.g., when combined with EB or AMReX's overset mask), which further complicates the setup.

For these reasons, we believe neither of the two boundary condition types currently supported in WarpX's electrostatic solver can represent the vanishing-potential-at-infinity condition in a satisfactory way.
As a side note, I have already validated this Robin-type boundary condition in a standalone AMReX program, and the results show non-negligible differences compared to the fully grounded case. However, this implementation was done with AMReX's cell-centered solver, as mentioned in my previous comment — while WarpX's electrostatic solver is nodal. Therefore, any guidance on how to bridge WarpX's nodal data layout with the cell-centered AMReX solver (specifically for the lab-frame electrostatic solver) would be greatly appreciated.
Thanks again for your time!

[WeiqunZhang]

We could add support for Robin BC in amrex's nodal Poisson solver, amrex::MLEBNodeFDLaplacian. Could you submit a request for feature in amrex github repo?