Question about EMF Corrections and CT

[pdmullen]

Hi, everyone! I have been thinking about EMF corrections recently and wanted to run a question by everyone:

For uniform meshes (without any refinement), Athena++ communicates EMFs between neighboring MeshBlocks and averages them. This is because when computing the EMF at boundaries for two different neighboring MeshBlocks, round-off error level differences can arise. My question, however, is…

After communicating and averaging EMF’s at MeshBlock boundaries, those updated EMF’s are used to perform CT. Can CT also produce round-off error level differences in the updated magnetic field at a face that is shared by two MeshBlock boundaries? If so, I worry that in highly chaotic simulations, the magnetic field value at the same face but from the perspective of two different MeshBlocks may begin to differ. Is this possible?

If this was an issue (which I am not convinced it is), the solution would be to just communicate the magnetic field component at the face shared by two MeshBlocks and average them (just like the EMFs). The resultant averaged magnetic field should still be divergence-less to round-off error.

[zhuzh1983]

I think the problem is that without doing so, EMF at the edges will never communicate between two different cores and their difference can grow in time. Eventually, the difference grows exponentially and is so large that the code crashed. Theoretically we don't have to do it at any step. For B and CT, if we only have a little bit round-off error at one step, this may not be an issue. If we just average B without correcting EMF, the EMF error can grow exponentially.

[tomidakn]

No, we do not send/receive the active surface B-field that is shared by two MeshBlocks. This is because manipulating the active B-field causes violation of divB=0.

[zhuzh1983]

Then, following the same logic, we shouldn't copy and average B1?

[tomidakn]

Well, as far as the error remains as small as the machine epsilon, it should be fine. But that's why I implemented only EMF correction and did not touch the face field itself.

[pdmullen]

No, we do not send/receive the active surface B-field that is shared by two MeshBlocks. This is because manipulating the active B-field causes violation of divB=0.

The post-CT B-fields at neighboring same-level MeshBlocks shared faces should be equal to ~round-off error. Averaging them should still keep divB to zero to machine precision, right?

[tomidakn]

I guess so, but if the change by the averaging is larger than (or on the same order of) the error in the CT, it might cause a new problem.

Since I have never experienced this issue by myself, I am not sure if averaging really helps. I think we only can tell by experiments with a reproducible test case.

[tomidakn]

I guess it is theoretically possible that the error in the CT phase may cause mismatch in the shared face field, but I have never seen such a case in my experience, including turbulent protoplanetary disks in spherical polar coordinates.