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Note: for this to work you need to apply the same summation in a deterministic order for all operations, including all of CarpetX's prolongation operations (and all derivatives in Z4c, which could eg be done inside out instead of left-to-right for central derivatives and in the direction of beta for the advection ones).
This exists an unpublished branch of GRhydro that does that and a (public but not advertised) branch of Carpet: rhaas/SymmetricOperators in case you want to peek (similar branches exist for other thorns as well).
Does work indeed (I don't think there is a paper that explicitly shows the difference but I have a presentation on it that I used for a job interview once), but is lots of work and fragile of course (and unphysical since nature is not that symmetric) and one should include a controllable physical non-symmetry in the ID, really).
Implement symmetry preserving summations which take care of the associative property. See this paper for details: https://www.sciencedirect.com/science/article/pii/S0045793018308399
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