In this chapter we investigate the accuracy of the different models presented in Chapter 2, in straight wall and circular geometries. The test cases are idealized in the sense that they are based on the linearized shallow water equations and therefore, an analytical solution exists. We are interested in the effective truncation order and the computational cost for all schemes. These considerations are important for the choice of a numerical method to use in ocean modelling. Although this approach is very basic, we stress the fact that these comparative studies are rarely done and that little is known about the relative effectiveness and cost of each scheme. For the finite difference (FD) models in a circular geometry, we are particularly interested in the influence of the steps for a wind-driven circulation that occurs along the walls when the discretization axes do not coincide with the orientation of the walls. These steps may have a detrimental effect on the overall effective truncation order. In contrast, finite element (FE) and spectral element (SE) models have much less difficulty in discretizing complex boundaries. However, the use of irregular grids may decrease the effective truncation order of these models. We perform a convergence-with-resolution study for a non-linear problem in a square domain. In this case, the reference solution is given by the high-order spectral element (SE) method at a high resolution. For this problem, we also present results using the simple adaptive strategy introduced in the previous chapter for the discontinuous SE method. When a dynamical boundary condition has to be found, we tend to focus on slip boundary conditions. Otherwise, the fluid is assumed to be inviscid.
For circular or smooth geometries it is possible to use curvilinear grids for FD methods and, hence, avoid the occurrence of steps along the boundaries. Curvilinear grids can better fit irregular coastlines and can provide some variable resolution capabilities, such as implemented in the POM [Blumberg and Herring, 1987] and SPEM [Song and Haidvogel, 1994] models. However, some smoothing of the geometry is needed, since curvilinear grids cannot accommodate all bays and capes. This method is therefore of limited use, since it accommodates only the large scale features of the coastline. For a realistic representation of lateral boundaries, step-like features would still appear, although the total number of steps is reduced when compared to Cartesian grids. We do not consider the use of curvilinear grids in our discussion of FD methods due to its lack of generality, although this method might be adequate for smoothly varying boundaries.
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