Page 150 - Computational Fluid Dynamics for Engineers

136 4. Numerical Methods for Model Parabolic and Elliptic Equations

4-11. The high and low frequency errors can be identified by using the s-called
modal analysis. To demonstrate this, consider the central difference approxima-
tions to,
- u"{x) = f{x) 0 < x < 1 (P4.11.1)
subject to the boundary conditions

u(0) = i/(l) = 0 (P4.11.2)

(a) show that Eq. (P4.11.1) can be written as
2
— Ui-i -f 2ui — iij+i = Ax fi 1 < i < N — 1 (P4.11.3)
or in matrix vector form as
Au = f (P4.11.4)

Here the tridiagonal matrix A is defined by
2 - 1
- 1 2 - 1
A = D - (L -f U) (P4.11.5)
- 1 2 - 1
- 1 2
where D is the diagonal of A and — L and — U are the strictly lower and upper
triangular parts of A, respectively.
(b) Show that the eigenvalues of A are

kn
Afc(A) = 4 sin' Kk< N - 1 (P4.11.6a)
2N
and that the corresponding eigenvectors are
/C7T
sin N Wfc,l

km
Wjfe sin N Wk,i 1 < k< i V - 1 , (P4.11.6b)

fc(JV-l)7T
sin N
and that the relation between eigenvalues A& and eigenvector w& is

= (P4.11.6c)
Aw fe X kw k
The parameter k denotes the wave number. The eigenvectors with wave numbers
in the range 1 < k < N/2 are called low-frequency mode. The eigenvectors with
N/2 < k < N — 1 are called high-frequency mode (See Fig. P4.2).
Hint: Show that Eq. (P4.11.6) is equivalent to