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Ocean Modelling for Resource Characterization Chapter | 8 207

FIG. 8.11 An example of how a mesh is defined in an FEM model (here ADCIRC).

The previous equation is called the weak form of a differential equation, because
the integral (some kind of average) of L(u) is set to zero. Ω is the domain of the
problem and W j are the weight/test functions. Depending on the FEM scheme,
several types of weight functions are chosen. For instance, in the Galerkin
method, weight functions and basis functions are the same (W j = ψ i ).
After implementing Eq. (8.26), enough algebraic equations are formed to
compute the nodal values of the state variable.
FEM models usually use a simple file format to define the mesh. Fig. 8.11
shows a sample mesh file for the ADCIRC model. As this figure shows, first,
the total number of elements and nodes is written. Then, the coordinate of each
node in a coordinate system is defined. Each element in a triangular mesh is
uniquely defined by connecting three nodes. The state variables are computed at
each node (or centre of an element) and can be processed using postprocessing
software such as Matlab.

8.2.3 Finite Volume Method
Finite volume method (FVM), like FEM, is based on an unstructured (e.g.
triangular) mesh. Therefore, it is suitable for irregular and complex geometries.
FVM has another advantage over FEM for fluid mechanic problems. So far, the
numerical methods that we presented have been based on PDEs. By contrast,
FVM is based on the integral form of the conservation laws, rather than their
differential form. This leads to more accuracy/stability, especially for sharp
gradients (i.e. large derivatives) inside a domain, which is also called shock-
capturing property. To explain this more clearly, as we mentioned before,
the dynamics of flow can be described by conservation of mass, momentum,
and energy. These conservation laws can be written as a system of PDEs.