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7  %\end{rawhtml}  %\end{rawhtml}
8    
9  \subsection{Adams-Bashforth III}  \subsection{Adams-Bashforth III}
10  \begin{figure}  
11    \begin{figure}[ht]
12  \begin{center}  \begin{center}
13  \resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_oscil.eps}}  \resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_oscil.eps}}
14  \end{center}  \end{center}
15  \caption{  \caption{
16  Comparaison of the oscillatory response of Adams-Bashforth schemes:  Comparison of the oscillatory response of Adams-Bashforth scheme.
17  }  }
18  \label{fig:ab_oscill_response}  \label{fig:ab_oscill_response}
19  \end{figure}  \end{figure}
20    
21  \begin{figure}  The third-order Adams-Bashforth time stepping (AB-3) provides
22  \begin{center}  several advantages (see, e.g., \cite{durr:91}) compared to
23  \resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_dampR.eps}}  the default quasi-second order Adams-Bashforth (AB-2):
 \end{center}  
 \caption{  
 Comparaison of the damping (diffusion like) response of Adams-Bashforth schemes:  
 }  
 \label{fig:ab_damp_response}  
 \end{figure}  
   
 As seen on fig.\ref{fig:adams-bashforth-respons}  
 The third-order Adams-Bashforth time stepping (AB-3) can be used instead  
 of the default quasi-second order Adams-Bashforth (AB-2),  
 with several advantages (see, e.g., \cite{durr:91}):  
24  \begin{itemize}  \begin{itemize}
25  \item higher accuracy.  \item higher accuracy;
26  \item stable with a longer time-step (for an oscillatory problem  \item stable with a longer time-step;
27  like advection or corriolis, stable up to a CFL of 0.72,  \item no additional computation (just requires the storage of one additional
28  compared to only 0.50 with AB-2 and $\epsilon_{AB} = 0.1$)   time level).
 (fig.\ref{fig:ab_oscill_response})  
 \item no additional computation, but only requires to store one additional  
  time level.  
29  \end{itemize}  \end{itemize}
30    
31  The extrapolation forward in time of the tendency (replacing equation  The $3^{rd}$ order Adams-Bashforth can be used to
32  \ref{eq:adams-bashforth2} can be written:  extrapolate forward in time the tendency
33    (replacing equation \ref{eq:adams-bashforth2})
34    which writes:
35  \begin{equation}  \begin{equation}
36  G_\tau^{(n+1/2)} = ( 1 + \alpha_{AB} + \beta_{AB}) G_\tau^n  G_\tau^{(n+1/2)} = ( 1 + \alpha_{AB} + \beta_{AB}) G_\tau^n
37  - ( \alpha_{AB} - 2 \beta_{AB}) G_\tau^{n-1}  - ( \alpha_{AB} + 2 \beta_{AB}) G_\tau^{n-1}
38  + \beta_{AB}) G_\tau^{n-2}  + \beta_{AB} G_\tau^{n-2}
39  \label{eq:adams-bashforth3}  \label{eq:adams-bashforth3}
40  \end{equation}  \end{equation}
41  with $(\alpha_{AB},\beta_{AB}) = (1/2, 5/12)$ corresponding to the  The 3rd order AB is obtained
42  3rd order AB. One can also recover  with $(\alpha_{AB},\,\beta_{AB}) = (1/2,\,5/12)$.
43  The quasi-2nd order AB corresponds to the particular case  Note that selecting
44  $(\alpha_{AB},\beta_{AB}) = (1/2+\epsilon_{AB}, 0)$.  $(\alpha_{AB},\,\beta_{AB}) = (1/2+\epsilon_{AB},\,0)$
45    one recovers the quasi-2nd order AB.
46  One can also extend the stability limit  %as illustrated on fig.\ref{fig:adams-bashforth-respons}.
47  up to a CFL of 0.786 for an oscillatory problem  
48    The AB-3 time stepping improves the stability limit
49    for an oscillatory problem like advection or Coriolis.
50    As seen from Fig.\ref{fig:ab_oscill_response},
51    it remains stable up to a CFL of 0.72,
52    compared to only 0.50 with AB-2 and $\epsilon_{AB} = 0.1$.
53    %
54    It is interesting to note that the stability limit can be further
55    extended up to a CFL of 0.786 for an oscillatory problem
56  (see fig.\ref{fig:ab_oscill_response})  (see fig.\ref{fig:ab_oscill_response})
57  using $(\alpha_{AB},\beta_{AB}) = (0.5, 0.2811)$  using $(\alpha_{AB},\,\beta_{AB}) = (0.5,\,0.2811)$
58  but then the scheme is only 2nd order accurate.  but then the scheme is only 2nd order accurate.
59    
60    \begin{figure}[ht]
61    \begin{center}
62    \resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_dampR.eps}}
63    \end{center}
64    \caption{
65    Comparison of the damping (diffusion like) response of Adams-Bashforth schemes.
66    }
67    \label{fig:ab_damp_response}
68    \end{figure}
69    
70  However, the behavior of the AB-3 for a damping problem (like diffusion)  However, the behavior of the AB-3 for a damping problem (like diffusion)
71  is less favorable, since the stability limit is reduced to  is less favorable, since the stability limit is reduced to
72  0.54 only (and 0.64 with $\beta_{AB} = 0.2811$) compared to 1. (and 0.9  0.54 only (and 0.64 with $\beta_{AB} = 0.2811$) compared to 1. (and 0.9
73  with $\epsilon_{AB} = 0.1$) with the AB-2 (see fig.\ref{fig:ab_damp_response}).  with $\epsilon_{AB} = 0.1$) with the AB-2 (see fig.\ref{fig:ab_damp_response}).
74    
75  A way to enable the use of a longer time step is  A way to enable the use of a longer time step is
76  to keep the dissipation terms ouside the AB extrapolation  to keep the dissipation terms outside the AB extrapolation
77  (therefore using a simple forward time-stepping) (setting  (setting {\em momDissip\_In\_AB=.FALSE.} in main parameter file
78  momDissip\_In\_AB=.FALSE. in main parameter file "data",  "\texttt{data}", namelist {\em PARM03}),
79  namelist PARM03), and use AB-3 for advection and corriolis terms.  thus returning to a simple forward time-stepping for dissipation,
80    and to use AB-3 only for advection and Coriolis terms.
81    
82  The AB-3 time stepping is activated by defining the option  The AB-3 time stepping is activated by defining the option
83  \#define ALLOW\_ADAMSBASHFORTH\_3  {\em \#define ALLOW\_ADAMSBASHFORTH\_3}
84  in CPP\_OPTIONS.h  in "\texttt{CPP\_OPTIONS.h}".
85  The parameters $\alpha_{AB},\beta_{AB}$ can be set from the  The parameters $\alpha_{AB},\beta_{AB}$ can be set from the
86  main parameter file "data" (namelist "PARM03") and their  main parameter file "\texttt{data}" (namelist {\em PARM03}) and their
87  default values correspond to the 3rd order Adams-Bashforth.  default value corresponds to the 3rd order Adams-Bashforth.
88  A simple example is provided in verification/advect\_xy/input.ab3\_c4.  A simple example is provided in "\texttt{verification/advect\_xy/input.ab3\_c4}".
89    
90  The AB-3 is not yet available for  The AB-3 is not yet available for
91  the vertical momentum equation (Non-Hydrostatic) and passive  the vertical momentum equation (Non-Hydrostatic)
92  tracers.  neither for passive tracers.
93    
94  \subsection{tracer rather than tendency time-extrapolation}  \subsection{Time-extrapolation of tracer (rather than tendency)}
95     (to be continued ...)
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