Comments on ‘A finite‐element scheme for the vertical discretization in the semi‐Lagrangian version of the ECMWF forecast model’ by A. Untch and M. Hortal (April B, 2004, 130, 1505–1530)

Aspects of the dynamical core of a nonhydrostatic, deep-atmosphere, unified weather and climate-prediction model

“…It is observed in [54] that: (a) the empirically estimated truncation errors of the linear FE and cubic collocation schemes are both of fourth order; (b) the measured errors of these two schemes are not only of the same order but identical (to the two significant figures given); and (c) the empirically estimated truncation error of the cubic FE scheme is of eighth order. In addition to explaining these results, the analysis of [55] shows that: (d) the truncation errors of the linear FE and cubic FE schemes applied to the integral form of the equations are respectively four and eight times smaller than those obtained by applying FE's to the differential form; and (e) the cubic FE scheme is formally equivalent for uniform resolution to a new ''heptic collocation'' scheme, in which a seventh-order spline is analytically integrated.…”

“…Recently, a new high-order finite-element (FE) vertical discretization scheme has been proposed for a hydrostatic primitive equation model using a terrain-following pressure-based vertical coordinate and an unstaggered grid [54]. This motivated its mathematical analysis in [55].…”