Page 157 - Computational Fluid Dynamics for Engineers

144 5. Numerical Methods for Model Hyperbolic Equations

The characteristics of the system given by Eq. (5.1.4) can be determined
from the procedure given in [1]. This requires that the eigenvalues of the Ja-
cobian matrix A, Eq. (5.1.5), and given by Eq. (5.1.8) are determined with
three solutions for A from the differential equations of the characteristics. The
compatibility equations can then be obtained by writing Eq. (5.1.4) along the
1
characteristics with the left eigenvectors, [L ] determined from
T
[V ] [A-XiI\ =-0 (5.1.10a)

or
[l\lUi}[A-XiI}=0; « = l 2,3 (5.1.10b)
If we let
1 1 2 3
X- = [L L L f (5.1.11)
I) with X - 1 , we obtain

x- 1 dQ dQ- = 0 (5.1.12)
dt dx
and noting that
1 l
X ~- A = AX~ (5.1.13)
where A represents the eigenvalues of the Jacobian matrix A, that is,

Ai
A = A 2 (5.1.14)
As

we can also write Eq. (5.1.13) as
l
X~ AX = A (5.1.15)
Here X represents the right eigenvector matrix of A.
Assuming that the Jacobian matrix A is diagonalizable with real eigenvalues,
the compatibility equations can be obtained for the case where A is either
constant or nonlinear. In the former case, the left eigenvectors of X -1 are auto-
matically constant and Eq. (5.1.12) becomes

l
l
~dX~ Q ( dX~ AXX~ L <?
dt dx = 0 (5.1.16a)
-
or l l
~dX~ Q dX' Q
(5.1.16b)
dt dx
When A is nonlinear, we first determine a vector Q such that

X (5.1.17)
dQ