In summary, all numerical methods have their advantages and drawbacks. Traditional FD methods are of low order (usually, second order) and very easy to implement but may lack accuracy due to the presence of steps in irregular domains. FE methods discretize easily complex domains but are generally of low order and require the solution of a matrix problem. Moreover, they may lose one order in truncation errors if the mesh is too irregular (which often occurs for triangular meshes). By contrast, traditional implementations of FD methods in ocean models make use of regular grids. SE methods offer high accuracy in complex domains but at an unknown cost and seem to lose accuracy in presence of steps. Therefore, they require the smoothly curved boundaries that we introduce in Section 2.4.4. We also introduce a simple adaptive mesh strategy for the SE method. The mesh is refined or derefined when the local error is too large. The local error is estimated based on the jump in the solution between two adjacent elements. Hence, the SE model should be able to automatically increase the resolution in regions where the solution is under-resolved. This might be essential in order to resolve and follow local eddies or moving fronts. The next step is to investigate the effective truncation order and the cost function for all the models presented in this chapter.
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