s_under_dvlp/text/time_stepping_dvlp.tex

% $Header: /u/gcmpack/manual/part2/time_stepping.tex,v 1.26 2006/06/29 01:45:32 jmc Exp $
% $Name:  $

\section{Other Time-stepping Options}
%\begin{rawhtml}
%<!-- CMIREDIR:dvlp-time-stepping: -->
%\end{rawhtml}

\subsection{Adams-Bashforth III}
\begin{figure}
\begin{center}
\resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_oscil.eps}}
\end{center}
\caption{
Comparaison of the oscillatory response of Adams-Bashforth schemes:
}
\label{fig:ab_oscill_response}
\end{figure}

\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}

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}):
\begin{itemize}
\item higher accuracy.
\item stable with a longer time-step (for an oscillatory problem
like advection or corriolis, stable up to a CFL of 0.72, 
compared to only 0.50 with AB-2 and $\epsilon_{AB} = 0.1$)
(fig.\ref{fig:ab_oscill_response})
\item no additional computation, but only requires to store one additional 
 time level.
\end{itemize}

The extrapolation forward in time of the tendency (replacing equation 
\ref{eq:adams-bashforth2} can be written:
\begin{equation}
G_\tau^{(n+1/2)} = ( 1 + \alpha_{AB} + \beta_{AB}) G_\tau^n
- ( \alpha_{AB} - 2 \beta_{AB}) G_\tau^{n-1}
+ \beta_{AB}) G_\tau^{n-2}
\label{eq:adams-bashforth3}
\end{equation}
with $(\alpha_{AB},\beta_{AB}) = (1/2, 5/12)$ corresponding to the 
3rd order AB. One can also recover 
The quasi-2nd order AB corresponds to the particular case
$(\alpha_{AB},\beta_{AB}) = (1/2+\epsilon_{AB}, 0)$.

One can also extend the stability limit
up to a CFL of 0.786 for an oscillatory problem 
(see fig.\ref{fig:ab_oscill_response})
using $(\alpha_{AB},\beta_{AB}) = (0.5, 0.2811)$
but then the scheme is only 2nd order accurate.

However, the behavior of the AB-3 for a damping problem (like diffusion)
is less favorable, since the stability limit is reduced to 
0.54 only (and 0.64 with $\beta_{AB} = 0.2811$) compared to 1. (and 0.9 
with $\epsilon_{AB} = 0.1$) with the AB-2 (see fig.\ref{fig:ab_damp_response}).

A way to enable the use of a longer time step is
to keep the dissipation terms ouside the AB extrapolation
(therefore using a simple forward time-stepping) (setting
momDissip\_In\_AB=.FALSE. in main parameter file "data",
namelist PARM03), and use AB-3 for advection and corriolis terms.

The AB-3 time stepping is activated by defining the option
\#define ALLOW\_ADAMSBASHFORTH\_3
in CPP\_OPTIONS.h
The parameters $\alpha_{AB},\beta_{AB}$ can be set from the
main parameter file "data" (namelist "PARM03") and their 
default values correspond to the 3rd order Adams-Bashforth.
A simple example is provided in verification/advect\_xy/input.ab3\_c4.

The AB-3 is not yet available for 
the vertical momentum equation (Non-Hydrostatic) and passive
tracers.

\subsection{tracer rather than tendency time-extrapolation}
1	jmc	1.1	% $Header: /u/gcmpack/manual/part2/time_stepping.tex,v 1.26 2006/06/29 01:45:32 jmc Exp $
2			% $Name: $
3
4			\section{Other Time-stepping Options}
5			%\begin{rawhtml}
6			%<!-- CMIREDIR:dvlp-time-stepping: -->
7			%\end{rawhtml}
8
9			\subsection{Adams-Bashforth III}
10			\begin{figure}
11			\begin{center}
12			\resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_oscil.eps}}
13			\end{center}
14			\caption{
15			Comparaison of the oscillatory response of Adams-Bashforth schemes:
16			}
17			\label{fig:ab_oscill_response}
18			\end{figure}
19
20			\begin{figure}
21			\begin{center}
22			\resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_dampR.eps}}
23			\end{center}
24			\caption{
25			Comparaison of the damping (diffusion like) response of Adams-Bashforth schemes:
26			}
27			\label{fig:ab_damp_response}
28			\end{figure}
29
30			As seen on fig.\ref{fig:adams-bashforth-respons}
31			The third-order Adams-Bashforth time stepping (AB-3) can be used instead
32			of the default quasi-second order Adams-Bashforth (AB-2),
33			with several advantages (see, e.g., \cite{durr:91}):
34			\begin{itemize}
35			\item higher accuracy.
36			\item stable with a longer time-step (for an oscillatory problem
37			like advection or corriolis, stable up to a CFL of 0.72,
38			compared to only 0.50 with AB-2 and $\epsilon_{AB} = 0.1$)
39			(fig.\ref{fig:ab_oscill_response})
40			\item no additional computation, but only requires to store one additional
41			time level.
42			\end{itemize}
43
44			The extrapolation forward in time of the tendency (replacing equation
45			\ref{eq:adams-bashforth2} can be written:
46			\begin{equation}
47			G_\tau^{(n+1/2)} = ( 1 + \alpha_{AB} + \beta_{AB}) G_\tau^n
48			- ( \alpha_{AB} - 2 \beta_{AB}) G_\tau^{n-1}
49			+ \beta_{AB}) G_\tau^{n-2}
50			\label{eq:adams-bashforth3}
51			\end{equation}
52			with $(\alpha_{AB},\beta_{AB}) = (1/2, 5/12)$ corresponding to the
53			3rd order AB. One can also recover
54			The quasi-2nd order AB corresponds to the particular case
55			$(\alpha_{AB},\beta_{AB}) = (1/2+\epsilon_{AB}, 0)$.
56
57			One can also extend the stability limit
58			up to a CFL of 0.786 for an oscillatory problem
59			(see fig.\ref{fig:ab_oscill_response})
60			using $(\alpha_{AB},\beta_{AB}) = (0.5, 0.2811)$
61			but then the scheme is only 2nd order accurate.
62
63			However, the behavior of the AB-3 for a damping problem (like diffusion)
64			is less favorable, since the stability limit is reduced to
65			0.54 only (and 0.64 with $\beta_{AB} = 0.2811$) compared to 1. (and 0.9
66			with $\epsilon_{AB} = 0.1$) with the AB-2 (see fig.\ref{fig:ab_damp_response}).
67
68			A way to enable the use of a longer time step is
69			to keep the dissipation terms ouside the AB extrapolation
70			(therefore using a simple forward time-stepping) (setting
71			momDissip\_In\_AB=.FALSE. in main parameter file "data",
72			namelist PARM03), and use AB-3 for advection and corriolis terms.
73
74			The AB-3 time stepping is activated by defining the option
75			\#define ALLOW\_ADAMSBASHFORTH\_3
76			in CPP\_OPTIONS.h
77			The parameters $\alpha_{AB},\beta_{AB}$ can be set from the
78			main parameter file "data" (namelist "PARM03") and their
79			default values correspond to the 3rd order Adams-Bashforth.
80			A simple example is provided in verification/advect\_xy/input.ab3\_c4.
81
82			The AB-3 is not yet available for
83			the vertical momentum equation (Non-Hydrostatic) and passive
84			tracers.
85
86			\subsection{tracer rather than tendency time-extrapolation}