s_under_dvlp/text/time_stepping_dvlp.tex

% $Header: /u/gcmpack/manual/s_under_dvlp/text/time_stepping_dvlp.tex,v 1.3 2010/08/25 03:00:54 jmc Exp $
% $Name:  $

\section{Other Time-stepping Options}
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\subsection{Adams-Bashforth III}

\begin{figure}[ht]
\begin{center}
\resizebox{10cm}{!}{\includegraphics{s_under_dvlp/figs/stab_AB3_oscil.eps}}
\end{center}
\caption{
Comparison of the oscillatory response of Adams-Bashforth scheme.
}
\label{fig:ab_oscill_response}
\end{figure}

The third-order Adams-Bashforth time stepping (AB-3) provides
several advantages (see, e.g., \cite{durr:91}) compared to
the default quasi-second order Adams-Bashforth (AB-2):
\begin{itemize}
\item higher accuracy;
\item stable with a longer time-step;
\item no additional computation (just requires the storage of one additional 
 time level).
\end{itemize}

The $3^{rd}$ order Adams-Bashforth can be used to 
extrapolate forward in time the tendency
(replacing equation \ref{eq:adams-bashforth2}) 
which writes:
\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}
The 3rd order AB is obtained 
with $(\alpha_{AB},\,\beta_{AB}) = (1/2,\,5/12)$.
Note that selecting 
$(\alpha_{AB},\,\beta_{AB}) = (1/2+\epsilon_{AB},\,0)$
one recovers the quasi-2nd order AB.
%as illustrated on fig.\ref{fig:adams-bashforth-respons}.

The AB-3 time stepping improves the stability limit
for an oscillatory problem like advection or Coriolis. 
As seen from Fig.\ref{fig:ab_oscill_response},
it remains stable up to a CFL of 0.72, 
compared to only 0.50 with AB-2 and $\epsilon_{AB} = 0.1$.
%
It is interesting to note that the stability limit can be further
extended 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.

\begin{figure}[ht]
\begin{center}
\resizebox{10cm}{!}{\includegraphics{s_under_dvlp/figs/stab_AB3_dampR.eps}}
\end{center}
\caption{
Comparison of the damping (diffusion like) response of Adams-Bashforth schemes.
}
\label{fig:ab_damp_response}
\end{figure}

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 outside the AB extrapolation
(setting {\em momDissip\_In\_AB=.FALSE.} in main parameter file 
"\texttt{data}", namelist {\em PARM03}),
thus returning to a simple forward time-stepping for dissipation,
and to use AB-3 only for advection and Coriolis terms.

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

The AB-3 is not yet available for 
the vertical momentum equation (Non-Hydrostatic) 
neither for passive tracers.

\subsection{Time-extrapolation of tracer (rather than tendency)}
 (to be continued ...)
1	% $Header: /u/gcmpack/manual/s_under_dvlp/text/time_stepping_dvlp.tex,v 1.3 2010/08/25 03:00:54 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
11	\begin{figure}[ht]
12	\begin{center}
13	\resizebox{10cm}{!}{\includegraphics{s_under_dvlp/figs/stab_AB3_oscil.eps}}
14	\end{center}
15	\caption{
16	Comparison of the oscillatory response of Adams-Bashforth scheme.
17	}
18	\label{fig:ab_oscill_response}
19	\end{figure}
20
21	The third-order Adams-Bashforth time stepping (AB-3) provides
22	several advantages (see, e.g., \cite{durr:91}) compared to
23	the default quasi-second order Adams-Bashforth (AB-2):
24	\begin{itemize}
25	\item higher accuracy;
26	\item stable with a longer time-step;
27	\item no additional computation (just requires the storage of one additional
28	time level).
29	\end{itemize}
30
31	The $3^{rd}$ order Adams-Bashforth can be used to
32	extrapolate forward in time the tendency
33	(replacing equation \ref{eq:adams-bashforth2})
34	which writes:
35	\begin{equation}
36	G_\tau^{(n+1/2)} = ( 1 + \alpha_{AB} + \beta_{AB}) G_\tau^n
37	- ( \alpha_{AB} + 2 \beta_{AB}) G_\tau^{n-1}
38	+ \beta_{AB} G_\tau^{n-2}
39	\label{eq:adams-bashforth3}
40	\end{equation}
41	The 3rd order AB is obtained
42	with $(\alpha_{AB},\,\beta_{AB}) = (1/2,\,5/12)$.
43	Note that selecting
44	$(\alpha_{AB},\,\beta_{AB}) = (1/2+\epsilon_{AB},\,0)$
45	one recovers the quasi-2nd order AB.
46	%as illustrated on fig.\ref{fig:adams-bashforth-respons}.
47
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})
57	using $(\alpha_{AB},\,\beta_{AB}) = (0.5,\,0.2811)$
58	but then the scheme is only 2nd order accurate.
59
60	\begin{figure}[ht]
61	\begin{center}
62	\resizebox{10cm}{!}{\includegraphics{s_under_dvlp/figs/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)
71	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
73	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
76	to keep the dissipation terms outside the AB extrapolation
77	(setting {\em momDissip\_In\_AB=.FALSE.} in main parameter file
78	"\texttt{data}", namelist {\em PARM03}),
79	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
83	{\em \#define ALLOW\_ADAMSBASHFORTH\_3}
84	in "\texttt{CPP\_OPTIONS.h}".
85	The parameters $\alpha_{AB},\beta_{AB}$ can be set from the
86	main parameter file "\texttt{data}" (namelist {\em PARM03}) and their
87	default value corresponds to the 3rd order Adams-Bashforth.
88	A simple example is provided in "\texttt{verification/advect\_xy/input.ab3\_c4}".
89
90	The AB-3 is not yet available for
91	the vertical momentum equation (Non-Hydrostatic)
92	neither for passive tracers.
93
94	\subsection{Time-extrapolation of tracer (rather than tendency)}
95	(to be continued ...)