Comment (by Roland Haas):

Summarizing part of the discussion above (and offline between @Zach Etienne and myself), the current “midpoint” method is incorrect / inaccurate in various ways.

As far as I can tell, the closest one could get it to be correct, without changing the interpolation points, is to remove the double counting in “phi” which turns it into the “trapezoidal” rule. Hence changing the default to “trapezoid” is the least disruptive change before the release.

@Zach Etienne has a patch / pull request that will provide actual “midpoint” rule to be merged into the release branch after the release once it has had time to mature a bit on master.

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Ticket URL: https://bitbucket.org/einsteintoolkit/tickets/issues/2398/multipole-default-midpoint-integration