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Several timestepping schemes are considered and used in conjunction with
one of the spatial discretization techniques proposed in the following
sections. For clarity, we review the timestepping techniques separately
in this section. Let us consider the equation

(2.3) 
The timeoperator can be finitedifferenced using a Taylor's series expansion
truncated after the first term:

(2.4) 
The simplest time discretization consists then of integrating
(2.3) given the previous timestep fields. This formulation
corresponds to the socalled explicit forward Euler scheme and is only
first order accurate

(2.5) 
This formulation is usually recommended for the integration of the
dissipative or friction terms, because no large precision is required in
time, as long as the small scale numerical noise are damped (and the
scheme is stable). For ensuring stability, a condition on the
magnitude of the time step, ,
applies. For instance, when
(a viscous dissipation term), this condition is

(2.6) 
Unfortunately, the forward Euler scheme is not neutral for various
problems, in the sense that some quantities such as mass, momentum or
energy are not conserved but may decay or grow as the simulation is
advanced in time. When these quantities grow with time, the model is of
course unstable. This happens, for instance, when F(u) represents the
Coriolis terms. In order to better conserve certain quantities, a better
scheme is the explicit leapfrog scheme

(2.7) 
based on a second order truncation

(2.8) 
The leapfrog scheme is thus centered in time. As the Euler
scheme, this scheme is restricted to certain conditions for stability.
For instance, if F(u)represents an advection or a wave propagation problem and using the
definition that the Courant number is given by

(2.9) 
where c is a phase speed
or an advection velocity, the CFL
(CourantFriedrichLevy) condition implies that
for stability. The Leapfrog scheme is neutral and conditionally stable
for problems involving, for instance, Coriolis or nonlinear advection
terms, and is unstable for dissipative terms. Moreover, the leapfrog
scheme requires a timefiltering, because the nonlinearities and
roundoff errors lead to a decoupling of the solution between even and odd
time steps. In order to avoid restriction of timestep magnitude, other
timeintegrations techniques were introduced. They include implicit and
semiimplicit schemes. Nonetheless, these schemes have to respect a
certain condition on the Courant number for ensuring a good accuracy. The
semiimplicit ^{2.1} scheme
consists of

(2.11) 
where
F(u^{n+1/2}) = 1/2 ( F(u^{n+1}) +
F(u^{n}) ) and the fullyimplicit scheme (also called the backward
Euler scheme) is implemented as

(2.12) 
The advantage of the semiimplicit treatment is that the timeoperator is
centered and secondorder. For the Coriolis terms, the fullyimplicit is
dissipative, and the semiimplicit (centered in time) is neutral. For the
treatment of the fast linear gravity waves present in the shallow water
equations, the advantage of using a semiimplicit or fullyimplicit
technique is that there is no restriction on timesteps (the domain of
stability of the models is extended) but at the expense of solving a
matrix problem due to the coupling of the variables through partial
derivatives.
The disadvantage of
these two techniques is that some physical processes, such as gravity
waves, are slowed down if a too large timestep is used (i.e., C>1).
This may have consequences for the interactions of important dynamical
processes (the geostrophic and ageostrophic modes) and, therefore, this
may lead to a less accurate representation of the cascade of energy (as
mentioned in Bartello and Thomas, 1996).
The nonlinear advective terms,
,
require a
special treatment. When we consider the computation of
,
they can be computed using the previous time step as
.
Then, if the timeoperator is centered and leapfrog, the
nonlinear terms are neutrally treated, otherwise, they are offcentered
for the other timeintegration schemes and may be unstable or dissipative
depending on the time integration techniques. The nonlinear terms can be
treated implicitly as
or fully implicit
using an iterative procedure. Another way is to use an explicit
4th order AdamsBashforth formulation

(2.13) 
The scheme is offcentered. It requires saving
fields from several previous timesteps and the timestep is limited by a
CFL condition. Use of RungeKutta techniques is also possible, the fourth
order one having the advantage of good quadratic conserving properties,
such as for the energy. But RungeKutta techniques require substep time
integrations as in this 4th order example:

(2.14) 
The RungeKutta formulations are neutral for all phenomena, very
accurate, and require a CFL condition. AdamsBashforth formulations are
usually recommended for nonlinear integrations, but have the practical
disadvantage of requiring smaller timesteps than equivalent order
RungeKutta integrations, to the point that there is no definite advantage
of one technique over the other^{2.2}. Hereafter, we tend to use the 4th order
Runge Kutta integration because of its accuracy and because it does not
require any time filtering.
A completely different timestepping approach consists of using the
Lagrangian framework (the grid follows the particles) instead of the
Eulerian framework implicitly assumed previously (the grid is fixed in
time). The Lagrangian timeintegration takes advantage of the fact that
the dynamical equations are simplified when written in a Lagrangian form



(2.15) 



(2.16) 
where D_{t} is the Lagrangian or total time derivative.
This is another way of saying that the particle trajectory is the
characteristic line for the advectiveonly problem. Hence, the problematic
nonlinear terms appearing in the equations do not appear explicitly
(except for the term in the mass balance). The main difficulty is in
following the particles that form the flow, and especially expressing the
righthandside terms. In order to avoid this problem, the socalled
semiLagrangian formulation was developed which takes advantage of both
the Lagrangian and Eulerian frameworks (see Staniforth and Côté, 1991,
for a review). The righthandside terms are discretized on
the Eulerian framework (in which derivatives are easy to express) and the
timederivative is treated on the Lagrangian framework. The advantage is
in keeping a fixed grid or domain in time. An interpolation procedure is
used in order to transfer information from the Eulerian grid to the
Lagrangian grid (the particle trajectories). As the equations are
timestepped along the advective characteristic lines (the particle
trajectories), there are no limitations imposed by numerical stability on
the magnitude of the timestep due to the advective terms. Therefore, the
method is effective in advection dominated flows. To be precise, according
to Bartello and Thomas (1996), the method is effective only if the
spectrum of energy is very steep (not too much energy at the smallest
scales). Moreover, a semiimplicit or fullyimplicit method can be added
to the semiLagrangian treatment of the equations [Robert, 1981]. Thus
the model has virtually no limitations due to stability regarding
timestep magnitude with respect to any physical process described by the
momentum equations. However, the presence of orography is troublesome in
semiLagrangian methods, effectively reducing the allowable timestep
[Ritchie and Tanguay, 1996]. The advantage of using semiLagrangian methods in an
ocean where the topography is steep is, hence, unclear.
Since the equations are iterated in time, the interpolation can be very
damaging to the conservation properties of the flow (mass or energy). That
is why modelers have to use high order interpolation schemes (cubic or
more). Nonetheless, the interpolation technique is usually responsible for
a large numerical dissipation, difficult to minimize. On the other hand,
these models can run without explicit eddyviscosity or diffusivity.
Proponents of the semiLagrangian method never fail to mention that their
models run without explicit numerical viscosity, whereas opponents note
that semiLagrangian models offer no control over this implicit viscosity.
Another disadvantage of the semiLagrangian technique when coupled to the
semiimplicit or implicit method is related to the same argument against
the semiimplicit and implicit methods. Namely, that too large a timestep
distorts the physical processes and misrepresents the real cascade of
energy.
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Frederic Dupont
20010911