Common partial differential equations of computational hydraulics 47
first-order hyperbolic equation at point (i,n) is known as the Euler’s scheme
in Equation 3.34 and has the form
f
f
t
c
f
f
x
i
n
i
n
o
i
n
i
n
+
+
−
− +
−
(
) =
1
1
1
2
0
∆
∆
(3.33)
Rearranging for the unknown f i
n+1
, the explicit solution reads
f
f
c
t
x
f
f
i
n
i
n
o
i
n
i
n
+
+
−
= −
−
(
)
1
1
1
2
∆
∆
(3.34)
The resulting numerical scheme (Equation 3.34) is consistent, but the solution, for any combination of Δx and Δt values, is unstable. The instability is
the result of the fact that although the analytic solution depends only on the
preceeding values in space and time of the function f(x,t), the above scheme
in Equation 3.34 involves a preceding value in space that is f i
n
+1 .
3.3.3.2 Godunov’s numerical scheme
Using forward differences for the time derivative and backward differences
for the space derivative (forward in time, backward space [FTBS]), the
numerical scheme known as the Godunov’s upwind scheme reads
f
f
t
c
f
f
x
i
n
i
n
o
i
n
i
n
+
−
− +
−
(
) =
1
1
0
∆
∆
(3.35)
Δt
Characteristic lines (curves)
Time t
x-axis
f(x,t)
c o
f(x,t + Δt)
Δx = c o Δt
e signal f(x,t) propagates unchanged
Figure 3.11 Characteristic lines in the x-t plane.