s_under_dvlp/text/time_stepping_dvlp.tex

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\section{Other Time-stepping Options}
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\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	% $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}