# 1.3.5. Solution strategy¶

The method of solution employed in the **HPE**, **QH** and **NH** models
is summarized in Figure 1.19. Under all dynamics, a
2-d elliptic equation is first solved to find the surface pressure and
the hydrostatic pressure at any level computed from the weight of fluid
above. Under **HPE** and **QH** dynamics, the horizontal momentum
equations are then stepped forward and \(\dot{r}\) found from
continuity. Under **NH** dynamics a 3-d elliptic equation must be solved
for the non-hydrostatic pressure before stepping forward the horizontal
momentum equations; \(\dot{r}\) is found by stepping forward the
vertical momentum equation.

There is no penalty in implementing **QH** over **HPE** except, of
course, some complication that goes with the inclusion of
\(\cos \varphi \ \) Coriolis terms and the relaxation of the shallow
atmosphere approximation. But this leads to negligible increase in
computation. In **NH**, in contrast, one additional elliptic equation -
a three-dimensional one - must be inverted for \(p_{\rm nh}\). However
the ‘overhead’ of the **NH** model is essentially negligible in the
hydrostatic limit (see detailed discussion in Marshall et al. (1997) [MHPA97]
resulting in a non-hydrostatic algorithm that, in the hydrostatic limit,
is as computationally economic as the **HPEs**.