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}[ht]
\begin{center}
\resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_oscil.eps}}
\end{center}
\caption{
Comparison of the oscillatory response of Adams-Bashforth scheme.
}
\label{fig:ab_oscill_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 Coriolis, 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.

\begin{figure}[ht]
\begin{center}
\resizebox{10cm}{!}{\includegraphics{under_dvlp/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
(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 Coriolis 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{Time-extrapolation of tracer (rather than tendency)}
 (to be continued ...)
1	jmc	1.2	% $Header: /u/gcmpack/manual/under_dvlp/time_stepping_dvlp.tex,v 1.1 2008/01/17 22:36:10 jmc Exp $
2	jmc	1.1	% $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	jmc	1.2
11			\begin{figure}[ht]
12	jmc	1.1	\begin{center}
13			\resizebox{10cm}{!}{\includegraphics{under_dvlp/stab_AB3_oscil.eps}}
14			\end{center}
15			\caption{
16	jmc	1.2	Comparison of the oscillatory response of Adams-Bashforth scheme.
17	jmc	1.1	}
18			\label{fig:ab_oscill_response}
19			\end{figure}
20
21			As seen on fig.\ref{fig:adams-bashforth-respons}
22			The third-order Adams-Bashforth time stepping (AB-3) can be used instead
23			of the default quasi-second order Adams-Bashforth (AB-2),
24			with several advantages (see, e.g., \cite{durr:91}):
25			\begin{itemize}
26			\item higher accuracy.
27			\item stable with a longer time-step (for an oscillatory problem
28	jmc	1.2	like advection or Coriolis, stable up to a CFL of 0.72,
29	jmc	1.1	compared to only 0.50 with AB-2 and $\epsilon_{AB} = 0.1$)
30			(fig.\ref{fig:ab_oscill_response})
31			\item no additional computation, but only requires to store one additional
32			time level.
33			\end{itemize}
34
35			The extrapolation forward in time of the tendency (replacing equation
36			\ref{eq:adams-bashforth2} can be written:
37			\begin{equation}
38			G_\tau^{(n+1/2)} = ( 1 + \alpha_{AB} + \beta_{AB}) G_\tau^n
39	jmc	1.2	- ( \alpha_{AB} + 2 \beta_{AB}) G_\tau^{n-1}
40			+ \beta_{AB} G_\tau^{n-2}
41	jmc	1.1	\label{eq:adams-bashforth3}
42			\end{equation}
43			with $(\alpha_{AB},\beta_{AB}) = (1/2, 5/12)$ corresponding to the
44			3rd order AB. One can also recover
45			The quasi-2nd order AB corresponds to the particular case
46			$(\alpha_{AB},\beta_{AB}) = (1/2+\epsilon_{AB}, 0)$.
47
48			One can also extend the stability limit
49			up to a CFL of 0.786 for an oscillatory problem
50			(see fig.\ref{fig:ab_oscill_response})
51			using $(\alpha_{AB},\beta_{AB}) = (0.5, 0.2811)$
52			but then the scheme is only 2nd order accurate.
53
54	jmc	1.2	\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	jmc	1.1	However, the behavior of the AB-3 for a damping problem (like diffusion)
65			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
67			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
70	jmc	1.2	to keep the dissipation terms outside the AB extrapolation
71	jmc	1.1	(therefore using a simple forward time-stepping) (setting
72			momDissip\_In\_AB=.FALSE. in main parameter file "data",
73	jmc	1.2	namelist PARM03), and use AB-3 for advection and Coriolis terms.
74	jmc	1.1
75			The AB-3 time stepping is activated by defining the option
76			\#define ALLOW\_ADAMSBASHFORTH\_3
77			in CPP\_OPTIONS.h
78			The parameters $\alpha_{AB},\beta_{AB}$ can be set from the
79			main parameter file "data" (namelist "PARM03") and their
80			default values correspond to the 3rd order Adams-Bashforth.
81			A simple example is provided in verification/advect\_xy/input.ab3\_c4.
82
83			The AB-3 is not yet available for
84			the vertical momentum equation (Non-Hydrostatic) and passive
85			tracers.
86
87	jmc	1.2	\subsection{Time-extrapolation of tracer (rather than tendency)}
88			(to be continued ...)