Fractional step method

Solving the momentum equation \eqref{eq:momentumFV} 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 $\boldsymbol{u}^\star$ without worrying about satisfying the incompressibility constraint. We then project $\boldsymbol{u}^\star$ onto the space of divergence-free velocity fields to obtain a value for $\boldsymbol{u}^{n+1}$ that satisfies continuity.

We thus discretize the momentum equation as

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

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 \S\ref{sec:time-stepping}).

The projection is then performed

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

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

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

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.

We are currently investigating whether our projection method is indeed second-order accurate in both velocity and pressure (see \S\ref{sec:forced-flow}). However, it may not matter too much for simulating high Reynolds number geophysical fluids as David L. Brown , Ricardo Cortez , Michael L. Minion (2001) conclude that "Quite often, semi-implicit projection methods are applied to problems in which the viscosity is small. Since the predicted first-order errors in the pressure are scaled by $\nu$, it is not clear whether the improved pressure-update formula is beneficial in such situations. ... Finally, in some applications of projection methods, second-order accuracy in the pressure may not be relevant or in some cases even possible due to the treatment of other terms in the equations."