  Small-scale features of shallow water flow obtained from direct numerical
simulation (DNS) with two different computational codes for the shallow water
equations are gathered offline and subsequently employed with the aim of
constructing a reduced-order correction. This is used to facilitate
high-fidelity online flow predictions at much reduced costs on coarse meshes.
The resolved small-scale features at high resolution represent subgrid
properties for the coarse representation. Measurements of the subgrid dynamics
are obtained as the difference between the evolution of a coarse grid solution
and the corresponding DNS result. The measurements are sensitive to the
particular numerical methods used for the simulation on coarse computational
grids and can be used to approximately correct the associated discretization
errors. The subgrid features are decomposed into empirical orthogonal functions
(EOFs), after which a corresponding correction term is constructed. By
increasing the number of EOFs in the approximation of the measured values the
correction term can in principle be made arbitrarily accurate. Both
computational methods investigated here show a significant decrease in the
simulation error already when applying the correction based on the dominant
EOFs only. The error reduction accounts for the particular discretization
errors that incur and are hence specific to the particular simulation method
that is adopted. This improvement is also observed for very coarse grids which
may be used for computational model reduction in geophysical and turbulent flow
problems.
