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2.5 Explicit time-stepping: Adams-Bashforth

In describing the the pressure method above we deferred describing the time discretization of the explicit terms. We have historically used the quasi-second order Adams-Bashforth method for all explicit terms in both the momentum and tracer equations. This is still the default mode of operation but it is now possible to use alternate schemes for tracers (see section 2.17).

Figure 2.3: Calling tree for the Adams-Bashforth time-stepping of temperature with implicit diffusion. (THERMODYNAMICS, ADAMS_BASHFORTH2)
% latex2html id marker 4625 \fbox{ \begin{minipage}{4.5in} \begin{tabbing} aaa ... ...FF \\lq $\tau^{(n+1)}$\ (\ref{eq:tau-n+1-implicit}) \end{tabbing} \end{minipage} }

In the previous sections, we summarized an explicit scheme as:

$\displaystyle \tau^{*} = \tau^{n} + \Delta t G_\tau^{(n+1/2)}$ (2.23)

where $ \tau$ could be any prognostic variable ($ u$ , $ v$ , $ \theta $ or $ S$ ) and $ \tau^*$ is an explicit estimate of $ \tau^{n+1}$ and would be exact if not for implicit-in-time terms. The parenthesis about $ n+1/2$ indicates that the term is explicit and extrapolated forward in time and for this we use the quasi-second order Adams-Bashforth method:

$\displaystyle G_\tau^{(n+1/2)} = ( 3/2 + \epsilon_{AB}) G_\tau^n - ( 1/2 + \epsilon_{AB}) G_\tau^{n-1}$ (2.24)

This is a linear extrapolation, forward in time, to $ t=(n+1/2+{\epsilon_{AB}})\Delta t$ . An extrapolation to the mid-point in time, $ t=(n+1/2)\Delta t$ , corresponding to $ \epsilon_{AB}=0$ , would be second order accurate but is weakly unstable for oscillatory terms. A small but finite value for $ \epsilon _{AB}$ stabilizes the method. Strictly speaking, damping terms such as diffusion and dissipation, and fixed terms (forcing), do not need to be inside the Adams-Bashforth extrapolation. However, in the current code, it is simpler to include these terms and this can be justified if the flow and forcing evolves smoothly. Problems can, and do, arise when forcing or motions are high frequency and this corresponds to a reduced stability compared to a simple forward time-stepping of such terms. The model offers the possibility to leave the tracer and momentum forcing terms and the dissipation terms outside the Adams-Bashforth extrapolation, by turning off the logical flags forcing_In_AB (parameter file data, namelist PARM01, default value = True). (tracForcingOutAB, default=0, momForcingOutAB, default=0) and momDissip_In_AB (parameter file data, namelist PARM01, default value = True). respectively.

A stability analysis for an oscillation equation should be given at this point.

A stability analysis for a relaxation equation should be given at this point.

Figure 2.4: Oscillatory and damping response of quasi-second order Adams-Bashforth scheme for different values of the $ \epsilon _{AB}$ parameter (0., 0.1, 0.25, from top to bottom) The analytical solution (in black), the physical mode (in blue) and the numerical mode (in red) are represented with a CFL step of 0.1. The left column represents the oscillatory response on the complex plane for CFL ranging from 0.1 up to 0.9. The right column represents the damping response amplitude (y-axis) function of the CFL (x-axis).
\resizebox{5.5in}{!}{\includegraphics{s_algorithm/figs/oscil+damp_AB2.eps}}

next up previous contents
Next: 2.6 Implicit time-stepping: backward Up: 2. Discretization and Algorithm Previous: 2.4 Pressure method with   Contents
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