NESTING OCEAN MODELS
131
and u ext can be in that case a correct approximation of u loc . However,
as mentioned previously, L ext and L loc generally differ, as well as the
forcing terms and the bathymetries. The quality of u ext is then lesser,
which will degrade the estimation of u loc . Moreover, since this approach
is only one-way, u loc never acts on u ext , and the external model cannot
be improved.
Usual two-way nesting.
An immediate possibility to address this
shortcoming is to add a feedback from the local model onto the external
one. Formulation (7) then becomes
L ext u ext = f ext in Ω ext ∪ Ω loc × [0, T ]
then L loc u loc = f loc in Ω loc × [0, T ]
Bu loc = Bu ext on Γ × [0, T ]
then
u ext = Hu loc in Ω loc × [0, T ]
(8)
where H is an update operator, mapping u loc from its time and space
grid onto the grid of the external model. This implies of course that the
external model is fully available, and that both models are run together
with on-line interaction. The update can be performed at each external
model timestep, or less frequently.
In this approach, the local solution has some influence onto the external one, the goal being to get closer to the target problem (5) without
having to modify the external model.
Full coupling.
As mentioned previously, the correct approach
should be to solve (5). However this implies first to modify the external
model by defining an open boundary on Γ in order to avoid overlapping
Ω loc , and also to find an interaction procedure that makes u loc and u ext
satisfy the regularity conditions on Γ. We will see in §4.2 how this can
be done. Such methods are quite recent, and are not yet disseminated
in the ocean and atmosphere modelling community.
2.3
A numerical example
Let us now illustrate the different preceding approaches in the very
simple case of a 1-D ordinary differential equation. The problem is:
−ν(x)u (x) + u(x) = sin nπx
x ∈]0, 1[
u(0) = u(1) = 0
(9)
The local domain we are interested in is Ω loc =]a, b[; hence Ω ext =
]0, a[∪]b, 1[. ν(x) is displayed on Figure 2a. It is equal to ν 0 in Ω ext , and
131
and u ext can be in that case a correct approximation of u loc . However,
as mentioned previously, L ext and L loc generally differ, as well as the
forcing terms and the bathymetries. The quality of u ext is then lesser,
which will degrade the estimation of u loc . Moreover, since this approach
is only one-way, u loc never acts on u ext , and the external model cannot
be improved.
Usual two-way nesting.
An immediate possibility to address this
shortcoming is to add a feedback from the local model onto the external
one. Formulation (7) then becomes
L ext u ext = f ext in Ω ext ∪ Ω loc × [0, T ]
then L loc u loc = f loc in Ω loc × [0, T ]
Bu loc = Bu ext on Γ × [0, T ]
then
u ext = Hu loc in Ω loc × [0, T ]
(8)
where H is an update operator, mapping u loc from its time and space
grid onto the grid of the external model. This implies of course that the
external model is fully available, and that both models are run together
with on-line interaction. The update can be performed at each external
model timestep, or less frequently.
In this approach, the local solution has some influence onto the external one, the goal being to get closer to the target problem (5) without
having to modify the external model.
Full coupling.
As mentioned previously, the correct approach
should be to solve (5). However this implies first to modify the external
model by defining an open boundary on Γ in order to avoid overlapping
Ω loc , and also to find an interaction procedure that makes u loc and u ext
satisfy the regularity conditions on Γ. We will see in §4.2 how this can
be done. Such methods are quite recent, and are not yet disseminated
in the ocean and atmosphere modelling community.
2.3
A numerical example
Let us now illustrate the different preceding approaches in the very
simple case of a 1-D ordinary differential equation. The problem is:
−ν(x)u (x) + u(x) = sin nπx
x ∈]0, 1[
u(0) = u(1) = 0
(9)
The local domain we are interested in is Ω loc =]a, b[; hence Ω ext =
]0, a[∪]b, 1[. ν(x) is displayed on Figure 2a. It is equal to ν 0 in Ω ext , and