For periodic scalar upwind advection, the parent-average history has exact rank P+(P-1)min{L,r-1}; a flux-divergence queue attains this bound, and small Courant numbers make deep delayed information numerically unrecoverable.
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Exact Finite-Horizon Memory, Conditioning, and Dissipative Decay in Coarse Upwind Finite-Volume Prediction
For periodic scalar upwind advection, the parent-average history has exact rank P+(P-1)min{L,r-1}; a flux-divergence queue attains this bound, and small Courant numbers make deep delayed information numerically unrecoverable.