We have tested in this chapter different FD, FE and SE methods. We first rule out the possibility of using a high order A-grid FD model because, in the presence of irregular geometries and for an inviscid flow, the effective truncation order is less than second order accurate. This was demonstrated in a circular domain and is due to the presence of steps occurring along the boundary when discretizing complex domains on Cartesian grids. The order of the model may have been preserved in curvilinear geometry, but we did not consider curvilinear grids as they are limited to smooth domains. The same applies to the second order C-grid model, although the loss of accuracy is less severe.

We also considered FE methods, some of which are quite simple (equal-order
formulation). They all use linear basis functions for velocity and
therefore we expect these methods to be no more than second order
accurate. In fact, for linear applications in rectangular domains, the
effective truncation order of FE models is fairly close to two. There is
an increase in the errors due to the use of unstructured grids. This
increase is sufficient for FD methods to outperform FE methods in terms of
cost. On the other hand, in a circular domains the order of the FD methods
is closer to one than two. Thus to obtain the same accuracy, the cost of
using FD methods in irregular domains becomes quickly prohibitive with
increasing resolution compared to FE methods. However, for nonlinear
applications, all equal-order FE methods tend to be more dissipative,
mostly because of the stabilizing formulations that guarantee the
stability of the model. Hence, applications of these methods for
non-linear oceanic flows seems problematic. There are other FE methods
which are stable by construction, complying with the so-called LBB
condition, and are non-dissipative (see Section 2.3 for a review on
FE model stability issues). Unfortunately, the cost associated with these
models is fairly large (they generally use higher than linear basis
functions for the velocity and leads to fuller matrices). Moreover, as
these models use lower order basis functions for the elevation (or
pressure), the actual accuracy for this variable may be smaller compared
to other numerical methods. Since modern altimetry offers near global
coverage of the elevation of the oceans, a good FE ocean modelling
strategy may be to not sacrifice the accuracy for this variable. We used
the LLS model which fulfills the LBB stability condition as an
illustration. We showed that the velocity errors for a linear test case
are less for this model than those of the equal-order FE models and that
the elevation errors are greater. However, the increased accuracy in
velocity is exactly traded off by an increased cost. Unfortunately, the
nonlinear (original version) LLS model uses a semi-implicit
semi-Lagrangian time formulation, which leads to dissipation when applied
to the nonlinear Munk problem. Hence, all FE models considered are too
dissipative for nonlinear applications. We also investigate some of the
influence of the ``lumping'' of the mass matrix in FE models. Some authors
have stressed a loss in accuracy due to lumped mass matrices
[Gresho *et al. , *1978]. We found that the use of mass lumping has a detrimental
influence on the double-gyre experiments with the LW model. The structure of
the solution tends to be more realistic when no lumping of the mass matrix
is performed.

We next considered a method based on discontinuous spectral elements. The
SE method introduced in Chapter 2 shows a better accuracy than FE and FD
models for ** n_{c}>3**. The convergence orders are not optimal though and vary
between

The C-grid FD model using the vorticity-divergence stress tensor and the enstrophy conserving advective scheme might be a good candidate for general ocean modelling. The loss of accuracy of second-order FD methods in presence of step-like geometry is less than one order. This loss is less compared to that suffered by the 4th order FD model. A second order FD model might thus still be competitive compared to intricate LBB-complying FE methods. However there are other limitations. From Chapter 1 we know that FD methods have problems representing the fast Kelvin modes if the resolution is too low compared to the radius of deformation. Therefore, a FD model should have many points resolving the radius of deformation, which significantly increases the cost. However, in the context of the Munk problem, it is not clear how retarded Kelvin waves affect the steady state of the ocean. We propose to further investigate these issues in the context of the single gyre Munk problem in Chapter 4.

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