Time-stepping Schemes Review
<wikitex> Numerical time stepping uses a discrete approximation to: $$ \frac{\partial \phi(t)}{\partial t} = {\cal F}(t) \eqno{(1)} $$ where $\phi$ represents one of $u$, $v$, $C$, or $\zeta$ and ${\cal F}(t)$ represents all the right-hand-side terms. In ROMS, the goal is to find time-stepping schemes which are accurate where they are valid and damping on unresolved signals ( Shchepetkin and McWilliams (2008b)). Also, the preference is for time-stepping schemes requiring only one set of the right-hand-side terms so that different time-stepping schemes can be used for different terms in the equations. Finally, as mentioned in Table Timestep, not all versions of ROMS use the same time-stepping algorithm. We list some timestepping schemes here which are used or have been used by the ROMS/SCRUM family of models, plus a few to help explain some of the more esoteric ones.
Euler
The simplest approximation is the Euler time step: $$ \frac{\phi(t + \Delta t) - \phi(t)}{\Delta t} = {\cal F}(t) \eqno{(2)}$$ where you predict the next $\phi$ value based only on the current fields. This method is accurate to first order in $\Delta t$; however, it is unconditionally unstable with respect to advection.
Leapfrog
The leapfrog time step is accurate to O($\Delta t^2$): $$ \frac{\phi(t + \Delta t) - \phi(t - \Delta t)}{2\Delta t} = {\cal F}(t). \eqno{(3)}$$ This time step is more accurate, but it is unconditionally unstable with respect to diffusion. Also, the even and odd time steps tend to diverge in a computational mode. This computational mode can be damped by taking correction steps. SCRUM's time step on the depth-integrated equations was a leapfrog step with a trapezoidal correction (LF-TR) on every step, which uses a leapfrog step to obtain an initial guess of $\phi(t+\Delta t)$. We will call the right-hand-side terms calculated from this initial guess ${\cal F}^*(t+\Delta t)$: $$ \frac{\phi(t + \Delta t) - \phi(t)}{\Delta t} = \frac{1}{2}
\left[ {\cal F}(t) + {\cal F}^*(t+\Delta t) \right] . \eqno{(4)}$$
This leapfrog-trapezoidal time step is stable with respect to diffusion and it strongly damps the computational mode. However, the right-hand-side terms are computed twice per time step.
Third-order Adams-Bashforth (AB3)
The time step on SCRUM's full 3-D fields is done with a third-order Adams-Bashforth step. It uses three time-levels of the right-hand-side terms: $$ \frac{\phi(t + \Delta t) - \phi(t)}{\Delta t} =
\alpha {\cal F}(t) +
\beta {\cal F}(t - \Delta t) +
\gamma {\cal F}(t - 2 \Delta t) \eqno{(5)}$$
where the coefficients $\alpha$, $\beta$ and $\gamma$ are chosen to obtain a third-order estimate of $\phi(t + \Delta t)$. We use a Taylor series expansion: $$ \frac{\phi(t + \Delta t) - \phi(t)}{\Delta t} =
\phi^{\prime} + \frac{\Delta t}{2} \phi^{\prime \prime} +
\frac{\Delta t^2}{6} \phi^{\prime \prime \prime} + \cdots \eqno{(6})$$
where $$ \eqalign{{\cal F}(t) & = & \phi^{\prime} \cr
{\cal F}(t - \Delta t) & = & \phi^{\prime}
- \Delta t \phi^{\prime \prime}
+ \frac{\Delta t^2}{2} \phi^{\prime \prime \prime} + \cdots \cr
{\cal F}(t - 2\Delta t) & = & \phi^{\prime}
- 2\Delta t \phi^{\prime \prime}
+ 2\Delta t^2 \phi^{\prime \prime \prime} + \cdots } $$
We find that the coefficients are: $$
\alpha = \frac{23}{12}, \qquad
\beta = - \frac{4}{3}, \qquad
\gamma = \frac{5}{12}
$$ This requires one time level for the physical fields and three time levels of the right-hand-side information and requires special treatment on startup.
Forward-Backward
In equation (1) above, we assume that multiple equations for any number of variables are time stepped synchronously. For coupled equations, we can actually do better by time stepping asynchronously. Consider these equations: $$ \eqalign{\frac{\partial \zeta}{\partial t} &= {\cal F}(u) \cr \frac{\partial u}{\partial t} &= {\cal G}(\zeta) } \eqno{(7)} $$ If we time step them alternately, we can always be using the newest information: $$ \eqalign{\zeta^{n+1} &= \zeta^n + {\cal F}(u^n) \Delta t \cr
u^{n+1} &= u^n + {\cal G}(\zeta^{n+1}) \Delta t} \eqno{(8)} $$
This scheme is second-order accurate and is stable for longer time steps than many other schemes. It is however unstable for the advection term.
Forward-Backward Feedback (RK2-FB)
One option for solving equation (7) is a predictor-corrector with predictor step: $$ \eqalign{\zeta^{n+1,\star} &= \zeta^n + {\cal F}(u^n)\Delta t \cr
u^{n+1,\star} &= u^n + \left[\beta {\cal G}(\zeta^{n+1,\star}) +
(1-\beta) {\cal G}(\zeta^n)\right] \Delta t} \eqno{(9)} $$
and corrector step: $$ \eqalign{\zeta^{n+1} &= \zeta^n + \frac{1}{2} \left[{\cal F}(u^{n+1,\star}) +
{\cal F}(u^n) \right] \Delta t \cr
u^{n+1} &= u^n + \frac{1}{2} \left[\epsilon {\cal G}(\zeta^{n+1}) +
(1-\epsilon){\cal G}(\zeta^{n+1,\star}) + {\cal G}(\zeta^n)
\right] \Delta t} \eqno{(10)} $$
Setting $\beta = \epsilon = 0$ in the above, it becomes a standard second order Runge-Kutta scheme, which is unstable for a non-dissipative system. Adding the $\beta$ and $\epsilon$ terms adds Forward-Backward feedback to this algorithm, and allows us to improve both its accuracy and stability. The choice of $\beta = 1/3$ and $\epsilon = 2/3$ leads to a stable third-order scheme.
