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

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

Steven A. Orszag, Moshe Israeli, Michel O. Deville (1986) and David L. Brown, Ricardo Cortez, Michael L. Minion (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.