Fractional step method

In some models (e.g., NonhydrostaticModel or HydrostaticFreeSurfaceModel) solving the momentum coupled with the continuity equation can be cumbersome so instead we employ a fractional step method. To approximate the solution of the coupled system we first solve an approximation to the discretized momentum equation for an intermediate velocity field $\boldsymbol{v}^\star$ without worrying about satisfying the incompressibility constraint. We then project $\boldsymbol{v}^\star$ onto the space of divergence-free velocity fields to obtain a value for $\boldsymbol{v}^{n+1}$ that satisfies continuity.

For example, for the NonhydrostaticModel, if we ignore the background velocity fields and the surface waves, we thus discretize the momentum equation as

\[ \frac{\boldsymbol{v}^\star - \boldsymbol{v}^n}{\Delta t} = - \left[ \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol{v} \right]^{n+\frac{1}{2}} - \boldsymbol{f} \times \boldsymbol{v}^{n+\frac{1}{2}} + \boldsymbol{\nabla} \boldsymbol{\cdot} \left ( \nu \boldsymbol{\nabla} \boldsymbol{v}^{n+\frac{1}{2}} \right ) + \boldsymbol{F}_{\boldsymbol{v}}^{n+\frac{1}{2}} \, ,\]

where the superscript $n + \frac{1}{2}$ indicates that these terms are evaluated at time step $n + \frac{1}{2}$, which we compute explicitly (see Time-stepping section).

The projection is then performed

\[ \boldsymbol{v}^{n+1} = \boldsymbol{v}^\star - \Delta t \, \boldsymbol{\nabla} p^{n+1} \, ,\]

to obtain a divergence-free velocity field $\boldsymbol{v}^{n+1}$. Here the projection is performed by solving an elliptic problem for the pressure $p^{n+1}$ with the boundary condition

\[ \boldsymbol{\hat{n}} \boldsymbol{\cdot} \boldsymbol{\nabla} p^{n+1} |_{\partial\Omega} = 0 \, .\]

Orszag *et al.* (1986) and Brown *et al.* (2001) raise an important issue regarding these fractional step methods, which is that "while the velocity can be reliably computed to second-order accuracy in time and space, the pressure is typically only first-order accurate in the $L_\infty$-norm." The numerical boundary conditions must be carefully accounted for to ensure the second-order accuracy promised by the fractional step methods.