Solution of the one-dimensional convection-diffusion equation by means of the Finite Element Method - Biomechanics: Concepts and Computation - page 266

1

0.9

0.8

0.7

0.6

v

0.5

0.4

0.3

0.2

0.1

0

0 0.2 0.4 0.6 0.8 1

x

Figure 15.1
Solution to steady 1-D convection-diffusion problem.

Assuming v and c constant, the dimensionless form may be written as
1

∂ ,

u ∗
∂ t ∗ +

u ∗
∂ x ∗ =

u ∗
∂ x ∗

Fo ∂

Pe ∂

∂
∂ x ∗

(15.8)

with the Fourier number given by

Fo = c

L 2 ,

(15.9)

and the Peclet number given by

vL
c

Pe = . (15.10)

The Peclet number reflects the relative importance of convection compared to dif-
fusion. It will be demonstrated that with increasing Peclet number the numerical
solution of the convection-diffusion problem becomes more difficult.
In the next section we will start by studying the time discretization of an
instationary equation. After that in Section 15.4 the spatial discretization of the
convection-diffusion equation will be discussed.

15.3 Temporal discretization

Many algorithms have been developed for the temporal discretization of the
convection-diffusion equation. Only one method is discussed here: the so-called