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p
c=1
u( x = 0) = 0
u( x = 1) = 1


The convective velocity v will be varied. For v = 0, the diffusion limit, the solution
is obvious: u varies linearly in x from u = 0 at x = 0 to u = 1 at x = 1.
Figs. 15.3(a) to (d) show the solution for v = 1, 10, 25 and 100, respectively,
using a uniform element distribution with ten linear elements. For v = 1 and v =
10 the approximate solution uh (solid line) closely (but not exactly) follows the
exact solution (dashed line). However, for v = 25 the numerical solution starts to
demonstrate an oscillatory behaviour that is more prominent for v = 100. Careful
analysis of the discrete set of equations shows that the so-called element Peclet
number governs this oscillatory behaviour. The element Peclet number is de¬ned
as
vh
Peh = ,
2c
271 15.4 Spatial discretization

1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
u 0.5 u 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
x x
(a) (b)

1
1.2
0.8
1
0.6
0.8 0.4
0.6 0.2
u
u
0
0.4
“0.2
0.2
“0.4
0 “0.6
“0.2 “0.8
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
x
x
(c) (d)

Figure 15.3
Solution of the steady convection-diffusion equation for v = 1, 10, 25 and v = 100, respectively
using ten linear elements; solid line: approximate solution uh , dashed line: exact solution u.




where h is the element length. Above a certain critical value of Peh the solu-
tion behaves in an oscillatory fashion. To reduce possible oscillations the element
Peclet number should be reduced. For ¬xed v and c this can only be achieved by
reducing the element size h. For example, doubling the number of elements from
10 to 20 eliminates the oscillations at v = 25, see Fig. 15.4.
The oscillations that appear in the numerical solution of the steady convection-
diffusion equation may be examined as follows. Consider a domain that is
subdivided in two linear elements, each having a length equal to h. At one end
of the domain the solution is ¬xed to u = 0, while at the other end the solution
is set to u = 1, or any other arbitrary non-zero value. For constant v and c the
governing differential equation may be rewritten as

v du d2 u
’ 2 = 0.
c dx dx
The one-dimensional convection-diffusion equation
272

1

0.9

0.8

0.7

0.6

0.5
u


0.4

0.3

0.2

0.1

0
0 0.2 0.4 0.6 0.8 1
x

Figure 15.4
Solution of the steady convection-diffusion problem using 20 elements at v = 25.



The set of equations that results after discretization is, as usual:
Ku=f .
∼ ∼

If only two linear elements of equal length h are used the coef¬cient matrix K may
be written as
⎡ ¤ ⎡ ¤
’1 1 0 1 ’1 0
v⎢ ⎥ 1⎢ ⎥
K= ⎣ ’1 0 1 ¦ + ⎣ ’1 2 ’1 ¦ ,
2c h
0 ’1 1 0 ’1 1
where the ¬rst, asymmetric, part corresponds to the convective term and the sec-
ond, symmetric, part to the diffusion term. In the absence of a source term the
second component of f is zero. Let u1 and u3 be located at the ends of the domain

such that u1 = 0 and u3 = 1, then u2 is obtained from
vh
2u2 = 1 ’
.
2c
An oscillation becomes manifest if u2 < 0. To avoid this, the element Peclet
number should be smaller than one:
vh
Peh = < 1.
2c
Consequently, at a given convective velocity v and diffusion constant c, the mesh
size h can be chosen such that an oscillation-free solution results. In particular
for large values of v/c this may result in very ¬ne meshes. To avoid the use of
273 Exercises

t = 0.01 : 0.05 : 0.26
1

0.9

0.8

0.7

0.6

0.5
u




0.4

0.3

0.2

0.1

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x

Figure 15.5
Solution of the unsteady convection-diffusion problem using 20 elements at v = 10.



very ¬ne meshes, an alternative, stabilized formulation has been developed: the
so-called SUPG (Streamline-Upwind/Petrov“Galerkin) formulation. A discussion
of this method however, is beyond the scope of the present book.

Example 15.3 Let us consider the instationary convection-diffusion problem. For this prob-
lem the same outline as in the previous example for v = 10 is chosen and
we use a uniform distribution of 20 linear elements. The initial condition is
u( x, t = 0) = 0 throughout the domain. At the ¬rst time step the boundary
condition u( x = 1, t) = 1 is imposed. The unsteady solution is obtained using a
time step of t = 0.01, while θ = 0.5 is selected for the θ-scheme. Fig. 15.5
shows the time-evolution of the solution towards the steady state value (denoted
by the dashed line) for v = 10.



Exercises

15.1 Consider the domain = [ 0 1]. On the domain the one-dimensional steady
convection-diffusion equation:
d2 u
du
=c 2
v
dx dx
The one-dimensional convection-diffusion equation
274

holds. As boundary conditions, at x = 0, u = 0 and at x = 1, u = 1 are
speci¬ed.
(a) Prove that the exact solution is given by
1 v
u= ( 1 ’ e c x) .
v
1’e c

(b)Verify this by means of the script demo_fem1dcd, to solve the one-
dimensional convection-diffusion problem, which can be found in the
directory oned of the programme library mlfem_nac. Use ¬ve ele-
ments and select c = 1, while v is varied. Choose v = 0, v = 1,
v = 10 and v = 20. Explain the results.
(c) According to Section 15.4, the solution is expected to be oscillation
free if the element Peclet number is smaller than 1:
ah
Peh = < 1.
2c
Verify that this is indeed the case.
15.2 Investigate the unsteady convection-diffusion problem:
‚u ‚u ‚ ‚u
+v = c ,
‚x ‚x ‚x
dt
= [ 0 1] subject to the initial condition:
on the domain

uini ( x, t = 0) = 0,

inside the domain and the boundary conditions:

u = 0 at x = 0, u = 1 at x = 1.

The θ-scheme for time integration is applied. Modify the m-¬le
demo_fem1dcd accordingly. Use ten linear elements.
(a) Choose v = c = 1 and solve the problem with different values of
θ. Use θ = 0.5, θ = 0.4 and θ = 0.25. For each problem start
with a time step of 0.01 and increase the time step with 0.01 until a
maximum of 0.05. Describe what happens with the solution.
(b) In the steady state case, what is the maximum value of the convective
velocity v such that the solution is oscillation free for c = 1?
(c) Does the numerical solution remain oscillation free in the unsteady
case for θ = 0.5 and t = 0.001, 0.01, 0.1? What happens?
(d) What happens at t = 0.001, if the convective velocity is reduced?
15.3 Investigate the unsteady convection problem:
‚u ‚ ‚u
‚u
+v = c
‚t ‚x ‚x ‚x
275 Exercises

on the spatial domain = [ 0 1] and a temporal domain that spans t =
[ 0 0.5]. At x = 0 the boundary condition u = 0 is prescribed, while at x =
1, c du/dx = 0 is selected. The convection dominated case is investigated,
with v = 1 and c = 0.01. The problem is solved using the θ-scheme with
θ = 0.5, a time step t = 0.05 and 40 linear elements.
(a) First, the initial condition:
u(x, t = 0) = sin(2π x)
is considered. Adapt the demo_fem1dcd and the FEM program
accordingly. In particular, make sure that the initial condition is han-
dled properly in fem1dcd. (Hint: the initial condition is speci¬ed in
sol(:,1).) Solve this problem. Plot the solution for all time steps.
(b) Second, let the initial condition be given by
u( x, t = 0) = 0 for x < 0.25 and x > 0.5,

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