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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    
 \begin{figure}  
 \begin{center}  
 \resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_dampR.eps}}  
 \end{center}  
 \caption{  
 Comparaison of the damping (diffusion like) response of Adams-Bashforth schemes:  
 }  
 \label{fig:ab_damp_response}  
 \end{figure}  
   
21  As seen on fig.\ref{fig:adams-bashforth-respons}  As seen on fig.\ref{fig:adams-bashforth-respons}
22  The third-order Adams-Bashforth time stepping (AB-3) can be used instead  The third-order Adams-Bashforth time stepping (AB-3) can be used instead
23  of the default quasi-second order Adams-Bashforth (AB-2),  of the default quasi-second order Adams-Bashforth (AB-2),
# Line 34  with several advantages (see, e.g., \cit Line 25  with several advantages (see, e.g., \cit
25  \begin{itemize}  \begin{itemize}
26  \item higher accuracy.  \item higher accuracy.
27  \item stable with a longer time-step (for an oscillatory problem  \item stable with a longer time-step (for an oscillatory problem
28  like advection or corriolis, stable up to a CFL of 0.72,  like advection or Coriolis, stable up to a CFL of 0.72,
29  compared to only 0.50 with AB-2 and $\epsilon_{AB} = 0.1$)  compared to only 0.50 with AB-2 and $\epsilon_{AB} = 0.1$)
30  (fig.\ref{fig:ab_oscill_response})  (fig.\ref{fig:ab_oscill_response})
31  \item no additional computation, but only requires to store one additional  \item no additional computation, but only requires to store one additional
# Line 45  The extrapolation forward in time of the Line 36  The extrapolation forward in time of the
36  \ref{eq:adams-bashforth2} can be written:  \ref{eq:adams-bashforth2} can be written:
37  \begin{equation}  \begin{equation}
38  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
39  - ( \alpha_{AB} - 2 \beta_{AB}) G_\tau^{n-1}  - ( \alpha_{AB} + 2 \beta_{AB}) G_\tau^{n-1}
40  + \beta_{AB}) G_\tau^{n-2}  + \beta_{AB} G_\tau^{n-2}
41  \label{eq:adams-bashforth3}  \label{eq:adams-bashforth3}
42  \end{equation}  \end{equation}
43  with $(\alpha_{AB},\beta_{AB}) = (1/2, 5/12)$ corresponding to the  with $(\alpha_{AB},\beta_{AB}) = (1/2, 5/12)$ corresponding to the
# Line 60  up to a CFL of 0.786 for an oscillatory Line 51  up to a CFL of 0.786 for an oscillatory
51  using $(\alpha_{AB},\beta_{AB}) = (0.5, 0.2811)$  using $(\alpha_{AB},\beta_{AB}) = (0.5, 0.2811)$
52  but then the scheme is only 2nd order accurate.  but then the scheme is only 2nd order accurate.
53    
54    \begin{figure}[ht]
55    \begin{center}
56    \resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_dampR.eps}}
57    \end{center}
58    \caption{
59    Comparison of the damping (diffusion like) response of Adams-Bashforth schemes.
60    }
61    \label{fig:ab_damp_response}
62    \end{figure}
63    
64  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)
65  is less favorable, since the stability limit is reduced to  is less favorable, since the stability limit is reduced to
66  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
67  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}).
68    
69  A way to enable the use of a longer time step is  A way to enable the use of a longer time step is
70  to keep the dissipation terms ouside the AB extrapolation  to keep the dissipation terms outside the AB extrapolation
71  (therefore using a simple forward time-stepping) (setting  (therefore using a simple forward time-stepping) (setting
72  momDissip\_In\_AB=.FALSE. in main parameter file "data",  momDissip\_In\_AB=.FALSE. in main parameter file "data",
73  namelist PARM03), and use AB-3 for advection and corriolis terms.  namelist PARM03), and use AB-3 for advection and Coriolis terms.
74    
75  The AB-3 time stepping is activated by defining the option  The AB-3 time stepping is activated by defining the option
76  \#define ALLOW\_ADAMSBASHFORTH\_3  \#define ALLOW\_ADAMSBASHFORTH\_3
# Line 83  The AB-3 is not yet available for Line 84  The AB-3 is not yet available for
84  the vertical momentum equation (Non-Hydrostatic) and passive  the vertical momentum equation (Non-Hydrostatic) and passive
85  tracers.  tracers.
86    
87  \subsection{tracer rather than tendency time-extrapolation}  \subsection{Time-extrapolation of tracer (rather than tendency)}
88     (to be continued ...)
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