Page 215 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

204 Fundamentals of Ocean Renewable Energy

In this case, the truncation error is
2 2 3
Δx d f(x) (Δx) d f(x)
+ + ··· (8.13)
2! dx 2 3! dx 3
Because it involves terms in Δx and higher powers, we say that it is of order Δx,
written as O(Δx). Note that as Δx → 0 we obtain the true derivative.
The backward difference scheme given by
f(x) − f(x − Δx)
(8.14)
Δx
also has an error O(Δx).
A better approximation can be obtained using the central difference scheme
f(x + Δx) − f(x − Δx)
(8.15)
2Δx
From the Taylor series expansion we obtain
2
2
3
3
df(x) (Δx) d f(x) (Δx) d f(x)
f(x − Δx) = f(x) − Δx + − + ··· (8.16)
dx 2! dx 2 3! dx 3
Hence, by subtracting Eq. 8.16 from 8.11
3
2
f(x + Δx) − f(x − Δx) df (Δx) d f(x)
= + + ··· (8.17)
2Δx dx 3! dx 3
2
for the central difference scheme. In this case, the error is O(Δx ). Because Δx
2
is small, Δx <Δx. Therefore, a centred scheme has a smaller truncation error
(i.e. is more accurate) than a forward or backward scheme.
8.2.2 Finite Element Method

The majority of ocean modelling problems involve complex geometries such
as irregular coastlines, inlets, islands, and headlands. Regular (rectangular or
curvilinear) grids cannot conveniently resolve these complex geometries. Finite
element method (FEM) is based on an irregular (e.g. triangular) mesh that
can easily resolve complex geometries (e.g. Fig. 8.1B). FEM originated in the
area of solid mechanics to calculate stress and strain in structures. However,
it is today applied to a wide range of multiphysics problems, including fluid
mechanics and ocean modelling. FEM can be regarded as a general numerical
method to solve partial differential equations (PDEs). The implementation of
FEM to fluid/solid mechanics problems involves many steps, and a detailed
explanation of those steps is beyond the scope of this book. However, the basic
concepts of the method are introduced briefly, because many renewable energy
problems use FEM codes for numerical simulations. For instance, ADCIRC is
an FEM-based ocean circulation model that can be used to simulate the tidal
energy resource of a region (e.g. [5]). Also, FEM is a common technique for