Page 213 - Fundamentals of Ocean Renewable Energy Generating Electricity From The Sea

202 Fundamentals of Ocean Renewable Energy

where the variable u is the temperature. Because u(x) is a continuous function,
there are an infinite number of points along this rod. However, it is much easier
if we consider a finite number of discrete points along this rod. In numerical
notation
u(x i ) =? x = x 1 , x 2 , ... , x N (8.3)
If we distribute these discrete points uniformly along the rod, the spacing
between the grid points will be Δx = L/(N − 1). Consequently,
x 1 = 0, x 2 = Δx, x 3 = 2Δx, ... , x N = (N − 1)Δx (8.4)
To find the temperature at each point, we need to solve the following
differential equation
2
∂u ∂ u
= k (8.5)
∂t ∂x 2
where k is a constant and depends on the material property of the rod.
8.2 NUMERICAL METHODS

8.2.1 Finite Difference Method
To solve a differential equation like Eq. (8.5), we need to evaluate derivatives.
Recalling from calculus (Chapter 2), the derivative of a function is defined as:
du u(x + Δx) − u(x)
= lim (8.6)
dx Δx→0 Δx
From a geometrical perspective, the derivative at a point equals the slope of
the tangent line at this point.
As an approximation, the derivative can be estimated as
du u(x + Δx) − u(x) L
Δx = (8.7)
dx Δx N − 1
It is clear that as the number of grid points (N) increases, Eq. (8.7) leads to
more accurate values. Referring to the simple rod example, the derivative at a
general point x i can be approximated as

u(x i + Δx) − u(x i ) u(x i+1 ) − u(x i ) u(x i+1 ) − u(x i )
du
= = (8.8)
dx Δx Δx x i+1 − x i
x=x i
which is called a finite difference. Specifically, a finite difference of the form of
Eq. (8.8) is called a forward difference.
Alternatively, if we use function values at grid points x i−1 and x i , we call it
a backward difference:

u(x i ) − u(x i − Δx) u(x i ) − u(x i−1 ) u(x i ) − u(x i−1 )
du
= = (8.9)
dx Δx Δx x i − x i−1
x=x i