This paper focuses on the computational aspects of DO (a lot more approachable than the paper Chris linked which is very theoretical!). For me this was the best reference to understand DO:
https://epubs.siam.org/doi/abs/10.1137/16M1109394

This paper focuses on doing DO on flow problems (I personally find the previous paper more approachable but perhaps this helps too):
https://www.sciencedirect.com/science/article/pii/S0021999112005098

This paper is the conception of the DO equations and explains their derivation quite nicely:
https://www.sciencedirect.com/science/article/pii/S0167278909002917

If you pursue this, I already have some preliminary work on DO methods that you can build on. And it would be great to contribute this to
github.com/FHoltorf/LowRankIntegrators.jl

Dynamically Orthogonal (DO) Approximation #10

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