46 Computational Modelling in Hydraulic and Coastal Engineering
that needs to be satisfied, known as the Courant-Friedrichs-Levi (CFL) criterion, reads
c
t
x
o
∆
∆
≤ 1
(3.29)
A better insight of the numerical solution of the hyperbolic equations and
the accompanying numerical errors can be accomplished by the numerical solution of the first-order hyperbolic equation known as the advective
transport equation:
∂
∂
+
∂
∂
=
f
t
c
f
x
o
0
(3.30)
Considering that
c
dx
dt
o =
(3.31)
it is evident that the solution of Equation 3.31 is a family of straight lines
x = x o + c o t, with slope c o and x value at t = 0 equal to x = x o . Combining
Equations 3.30 and 3.31 yields
∂
∂
+
∂
∂
=
=
f
t
dx
dt
f
x
Df
Dt
0
(3.32)
where D/Dt is the material or total derivative of f(x,t). That form implies that
f(x,t) is constant along the characteristic curve c
dx
dt
o =
(see Equation 3.12).
As shown in Figure 3.11, the solution consists of a plain translation of the
initial form of f (initial condition f(x,t = 0)) along the positive x-axis direction with speed equal to c o .
Thus, from the physical point of view, Equation 3.30 describes the propagation with speed c o of a signal (wave) in the positive x-axis direction
only, and also the dependence of the function f(x,t) on the values of the
same function on preceeding space and time steps. Having the details of
the analytical solution, different numerical solution schemes can be utilized
and critically analysed.
3.3.3.1 Euler’s unstable numerical scheme
When the time derivative is approximated with a forward FD and the space
derivative with a central FD (FTCS), the algebraic approximation of the