Time-stepping Schemes Review - Revision history

Robertson at 13:10, 4 August 2015

2015-08-04T13:10:59Z

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Robertson at 18:21, 31 July 2015

2015-07-31T18:21:42Z

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Kate: /* Generalized FB with an AB3-AM4 Step */

2009-09-22T20:02:53Z

Generalized FB with an AB3-AM4 Step

← Older revision		Revision as of 20:02, 22 September 2009
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	$\gamma$, and $\epsilon$. It is third-order accurate if $\beta =		$\gamma$, and $\epsilon$. It is third-order accurate if $\beta =
	5/12$. However, it has a wider stability range for $\beta =		5/12$. However, it has a wider stability range for $\beta =
	0.281105$. [[Bibliography#ShchepetkinAF_2008b \| Shchepetkin and McWilliams (~~2008b~~)]] choose to use		0.281105$. [[Bibliography#ShchepetkinAF_2008b \| Shchepetkin and McWilliams (2009)]] choose to use
	this scheme for the barotropic mode in their model with $\beta=		this scheme for the barotropic mode in their model with $\beta=
	0.281105$, $\gamma = 0.0880$, and $\epsilon = 0.013$, to obtain		0.281105$, $\gamma = 0.0880$, and $\epsilon = 0.013$, to obtain

Kate at 20:02, 22 September 2009

2009-09-22T20:02:26Z

← Older revision		Revision as of 20:02, 22 September 2009
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	and ${\cal F}(t)$		and ${\cal F}(t)$
	represents all the right-hand-side terms. In ROMS, the goal is to find time-stepping schemes which are accurate		represents all the right-hand-side terms. In ROMS, the goal is to find time-stepping schemes which are accurate
	where they are valid and damping on unresolved signals ([[Bibliography#ShchepetkinAF_2005a \| Shchepetkin and McWilliams (2005)]] and [[Bibliography#ShchepetkinAF_2008b \| Shchepetkin and McWilliams (~~2008b~~)]]). Also, the preference is for time-stepping schemes requiring only one set of the right-hand-side terms so that different time-stepping schemes can be used for different terms in the equations. Finally, as mentioned in Table [[Numerical_Solution_Technique#Table_timestep1 \| Time Step]], not all versions of ROMS use the same time-stepping algorithm. We list some time-stepping schemes here which are used or have been used by the ROMS/SCRUM family of models, plus a few to help explain some of the more esoteric ones.		where they are valid and damping on unresolved signals ([[Bibliography#ShchepetkinAF_2005a \| Shchepetkin and McWilliams (2005)]] and [[Bibliography#ShchepetkinAF_2008b \| Shchepetkin and McWilliams (2009)]]). Also, the preference is for time-stepping schemes requiring only one set of the right-hand-side terms so that different time-stepping schemes can be used for different terms in the equations. Finally, as mentioned in Table [[Numerical_Solution_Technique#Table_timestep1 \| Time Step]], not all versions of ROMS use the same time-stepping algorithm. We list some time-stepping schemes here which are used or have been used by the ROMS/SCRUM family of models, plus a few to help explain some of the more esoteric ones.
	</wikitex>		</wikitex>
	== Euler ==		== Euler ==

Arango: Time-stepping Schemes moved to Time-stepping Schemes Review

2009-03-03T21:29:06Z

Time-stepping Schemes moved to Time-stepping Schemes Review

← Older revision	Revision as of 21:29, 3 March 2009
(No difference)

Arango at 21:28, 3 March 2009

2009-03-03T21:28:49Z

← Older revision		Revision as of 21:28, 3 March 2009
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	<div class="title">Time-stepping Schemes</div>		<div class="title">Time-stepping Schemes Review</div>
	<wikitex>		<wikitex>
	Numerical time stepping uses a discrete approximation to:		Numerical time stepping uses a discrete approximation to:

Kate: /* LF-TR and LF-AM3 with FB Feedback */

2008-09-04T18:36:12Z

LF-TR and LF-AM3 with FB Feedback

← Older revision		Revision as of 18:36, 4 September 2008
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	$$		$$
	\beta = \epsilon = 0 \Rightarrow \cases{		\beta = \epsilon = 0 \Rightarrow \cases{
	\gamma = 0 &$\Rightarrow \hbox{LF-TR}, \alpha_{\max} = \sqrt{2}$ \cr		\gamma = 0 &$\Rightarrow$ \hbox to 2.4cm{LF-TR, \hfil} $\alpha_{\max} = \sqrt{2}$ \cr
	\gamma = 1/12 &$\Rightarrow \hbox{LF-AM3}, \alpha_{\max} = 1.5874$ \cr		\gamma = 1/12 &$\Rightarrow$ \hbox to 2.4cm{LF-AM3, \hfil} $\alpha_{\max} = 1.5874$ \cr
	\gamma = 0.0804 &$\Rightarrow \hbox{max stability}, \alpha_{\max} = 1.5876$}		\gamma = 0.0804 &$\Rightarrow$ \hbox to 2.4cm{max stability, \hfil} $\alpha_{\max} = 1.5876$}
	$$		$$
	Keeping $\gamma = 1/12$ maintains third-order accuracy. The most		Keeping $\gamma = 1/12$ maintains third-order accuracy. The most
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	1.851640$.		1.851640$.
	</wikitex>		</wikitex>

	== Generalized FB with an AB3-AM4 Step ==		== Generalized FB with an AB3-AM4 Step ==
	<wikitex>One drawback of the previous two schemes is the need to evaluate the		<wikitex>One drawback of the previous two schemes is the need to evaluate the

Kate: done for now?

2008-09-04T18:29:06Z

done for now?

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Kate: First part, with errors, will fix tomorrow...

2008-09-04T00:46:10Z

First part, with errors, will fix tomorrow...

New page

<div class="title">Time-stepping Schemes</div>

<wikitex>
Numerical time stepping uses a discrete approximation to:
$$ \frac{\partial \phi(t)}{\partial t} = {\cal F}(t) \eqno{(1)} $$
where $\phi$ represents one of $u$, $v$, $C$, or $\zeta$
and ${\cal F}(t)$
represents all the right-hand-side terms. In ROMS, the goal is to find time-stepping schemes which are accurate
where they are valid and damping on unresolved signals ([[Bibliography#ShchepetkinAF_2008b | Shchepetkin and McWilliams (2008b)]]). Also, the preference is for time-stepping schemes requiring only one set of the right-hand-side terms so that different time-stepping schemes can be used for different terms in the equations. Finally, as mentioned in Table [[Numerical_Solution_Technique#Table_timestep1 | Timestep]], not all versions of ROMS use the same time-stepping algorithm. We list some timestepping schemes here which are used or have been used by the ROMS/SCRUM family of models, plus a few to help explain some of the more esoteric ones.

== Euler ==
The simplest approximation is the Euler time step:
$$ \frac{\phi(t + \Delta t) - \phi(t)}{\Delta t} = {\cal F}(t) \eqno{(2)}$$
where you predict the next $\phi$ value based only on the current
fields. This method is accurate to first order in $\Delta t$; however,
it is unconditionally unstable with respect to advection.

== Leapfrog ==
The leapfrog time step is accurate to O($\Delta t^2$):
$$ \frac{\phi(t + \Delta t) - \phi(t - \Delta t)}{2\Delta t} = {\cal F}(t). \eqno{(3)}$$
This time step is more accurate, but it is unconditionally unstable with respect to diffusion. Also, the even and odd time steps tend to diverge in a computational mode. This computational mode can be damped by taking correction steps. SCRUM's time step on the depth-integrated equations was a leapfrog step with a trapezoidal correction (LF-TR) on every step, which uses a leapfrog step to obtain an initial guess of $\phi(t+\Delta t)$. We will call the right-hand-side terms calculated from this initial guess ${\cal F}^*(t+\Delta t)$:
$$ \frac{\phi(t + \Delta t) - \phi(t)}{\Delta t} = \frac{1}{2}
\left[ {\cal F}(t) + {\cal F}^*(t+\Delta t) \right] . \eqno{(4)}$$
This leapfrog-trapezoidal time step is stable with respect to diffusion and it strongly damps the computational mode. However, the right-hand-side terms are computed twice per time step.

== Third-order Adams-Bashforth (AB3) ==
The time step on SCRUM's full 3-D fields is done with
a third-order Adams-Bashforth step. It uses three time-levels of the
right-hand-side terms:
$$ \frac{\phi(t + \Delta t) - \phi(t)}{\Delta t} =
\alpha {\cal F}(t) +
\beta {\cal F}(t - \Delta t) +
\gamma {\cal F}(t - 2 \Delta t) \eqno{(5)}$$
where the coefficients $\alpha$, $\beta$ and $\gamma$ are chosen to
obtain a third-order estimate of $\phi(t + \Delta t)$. We use a Taylor
series expansion:
$$ \frac{\phi(t + \Delta t) - \phi(t)}{\Delta t} =
\phi^{\prime} + \frac{\Delta t}{2} \phi^{\prime \prime} +
\frac{\Delta t^2}{6} \phi^{\prime \prime \prime} + \cdots \eqno{(6})$$
where
$$ \eqalign{{\cal F}(t) & = & \phi^{\prime} \cr
{\cal F}(t - \Delta t) & = & \phi^{\prime}
- \Delta t \phi^{\prime \prime}
+ \frac{\Delta t^2}{2} \phi^{\prime \prime \prime} + \cdots \cr
{\cal F}(t - 2\Delta t) & = & \phi^{\prime}
- 2\Delta t \phi^{\prime \prime}
+ 2\Delta t^2 \phi^{\prime \prime \prime} + \cdots } $$
We find that the coefficients are:
$$
\alpha = \frac{23}{12}, \qquad
\beta = - \frac{4}{3}, \qquad
\gamma = \frac{5}{12}
$$
This requires one time level for the physical fields and three time
levels of the right-hand-side information and requires special
treatment on startup.

== Forward-Backward ==
In equation (1) above, we assume that multiple equations
for any number of variables are time stepped synchronously. For
coupled equations, we can actually do better by time stepping
asynchronously. Consider these equations:
$$ \eqalign{\frac{\partial \zeta}{\partial t} &= {\cal F}(u) \cr \frac{\partial u}{\partial t} &= {\cal G}(\zeta) } \eqno{(7)} $$
If we time step them alternately, we can always be using the newest
information:
$$ \eqalign{\zeta^{n+1} &= \zeta^n + {\cal F}(u^n) \Delta t \cr
u^{n+1} &= u^n + {\cal G}(\zeta^{n+1}) \Delta t} \eqno{(8)} $$
This scheme is second-order accurate and is stable for longer
time steps than many other schemes. It is however unstable for the
advection term.

== Forward-Backward Feedback (RK2-FB) ==
One option for solving equation (7) is a predictor-corrector
with predictor step:
$$ \eqalign{\zeta^{n+1,\star} &= \zeta^n + {\cal F}(u^n)\Delta t \cr
u^{n+1,\star} &= u^n + \left[\beta {\cal G}(\zeta^{n+1,\star}) +
(1-\beta) {\cal G}(\zeta^n)\right] \Delta t} \eqno{(9)} $$
and corrector step:
$$ \eqalign{\zeta^{n+1} &= \zeta^n + \frac{1}{2} \left[{\cal F}(u^{n+1,\star}) +
{\cal F}(u^n) \right] \Delta t \cr
u^{n+1} &= u^n + \frac{1}{2} \left[\epsilon {\cal G}(\zeta^{n+1}) +
(1-\epsilon){\cal G}(\zeta^{n+1,\star}) + {\cal G}(\zeta^n)
\right] \Delta t} \eqno{(10)} $$
Setting $\beta = \epsilon = 0$ in the above, it becomes a standard
second order Runge-Kutta scheme, which is unstable for a
non-dissipative system. Adding the $\beta$ and $\epsilon$ terms adds
Forward-Backward feedback to this algorithm, and allows us to
improve both its accuracy and stability. The choice of $\beta = 1/3$
and $\epsilon = 2/3$ leads to a stable third-order scheme.