Two novel families of multiscale staggered patch schemes efficiently simulate large-scale, weakly damped, linear waves

The paper establishes, largely empirically, that (i) spectral coupling reproduces the microscale eigenvalues for all resolved macroscale modes to roundoff, (ii) polynomial coupling of order p yields eigenvalue errors that scale like Δ^p for fixed low wavenumbers, and (iii) the schemes are stable over wide parameter ranges. The model provides a coherent Fourier-based proof sketch for (i) and (ii) and a symbol-based stability rationale for (iii). One notable paper typo is the sign of the viscous contribution in the real (vortical) eigenvalue in Eq. (6); the physically consistent form is −(cD + cV ω0^2), as used by the model. Aside from this sign typo and the paper’s reliance on numerical evidence in lieu of formal proofs, the conclusions are consistent.