PDF file - Johannes Kepler University, Linz - JKU

CHAPTER 2. PRELIMINARIES 24
Table 2.1 Stability and accuracy of some time stepping methods.
A-stab. str. A-stab. L-stab. accuracy
explicit Euler no no no 1st order
implicit Euler yes yes yes 1st order
Crank-Nicolson yes no no 2nd order
frac.-step-θ yes yes no 2nd order
2.4 The Convective Term
The convective term (u∇)u resp. (w∇)u causes two problems we have to deal with. Firstly
instabilities may occur because of it, secondly we have to cope with its nonlinearity.
2.4.1 Instability
The unstable behavior can already be observed in the following 1D model problem.
Example 2.10. Assume that we want to solve the following scalar convection diffusion
equation for u:
−νu ′′ (x) + wu ′ (x) = f for x ∈ (0, 1),
u(0) = u(1) = 0, w, f and ν constant on [0, 1]. A linear finite elements discretization on a
regular grid with mesh-width h leads to the system
⎛
2ν
− ν + w ⎞ ⎛ ⎞ ⎛ ⎞
u
h h 2
1 fh
− ν − w 2ν
− ν + w h 2 h h 2
u 2
fh
. .. . .. . ..
.
=
.
.
⎜
⎝
. .. . .. . ⎟ ⎜ ⎟ ⎜ ⎟
.. ⎠ ⎝ . ⎠ ⎝ . ⎠
− ν − w 2ν
u
h 2 h n fh
} {{ }
=:A
The corresponding eigenvalue problem reads row-wise
( w
2 − ν h)
u i+1 +
( ) 2ν
(
h − λ u i + − w 2 − ν )
u i−1 = 0, for i = 1, . . . , n,
h
u 0 = u n+1 = 0,
where λ is the eigenvalue we are searching for. Assume that n is odd and wh ≠ 2ν, then
one solution can easily be found as λ = 2ν , u h 2k = 0, u 2k+1 = ( )
wh+2ν k,
wh−2ν for k = 0, . . . ,
n−1
. 2
Thus, for small ν this eigenvalue tends to zero and the very oscillatory eigenvector
(Figure 2.4) is amplified in the solution if h is not small enough.
A solution of this problem is to use a less centered discretization, test-functions with
more weight upstream than downstream. In the Streamline Upwinding Petrov Galerkin