iterative linear solvers in the context of conservation laws

https://link.springer.com/article/10.1007/s10543-023-00992-w

Q: Do iterative methods (in this case Newton) preserve nonlinear invariants? A: No. Q: What should one do about it? A: Relax the time step but leave the iterations alone.

https://link.springer.com/article/10.1007/s10915-022-01923-7

Q: Do iterative methods preserve basic properties of PDE discretizations such as conservation? A: Some do, some don’t. Q: So if we pick a conservative one, then all is fine? A: No, because consistency is lost (time is slowed down).

https://epubs.siam.org/doi/abs/10.1137/22M1503348

An extension of the previous paper. A bit harder to read but contains some new interesting results, not least on the connection between Krylov methods and pseudotime stepping.

https://onlinelibrary.wiley.com/doi/abs/10.1002/pamm.202200157

A deeper dive into the inconsistency (time dilation) caused by Krylov methods. I wrote this paper in about 48 hours so it misses a crucial simplification in the main result that I can tell you about if you want to know more.

Start by reading the four linked papers, especially the discussion of Newton, conservative iterative methods, Krylov methods, and pseudotime stepping. The issue does not name files, tests, entry points, or a concrete acceptance condition, so the first step is to clarify which solver behavior should change and how it should be validated.