156
3. Beyond the One-Way Wave Equation
yield a skew-symmetric matrix even when the velocities satisfy (3.107), and as a
consequence, it does not guarantee the conservation of lIull2.
As an alternative to the construction of A and the evaluation of its symmetry
properties, one can examine the stability and conservation properties of approximate solutions to (3.106) and (3.107) using the energy method to compute the
time tendency of the 12-norm of the serni-discrete solution from the relation
=!.. (",,!/>?t!..xt!.. ) =2""!/>. .d!/>i,j t!..xt!.. .
u't
dt
I ,)
Y
I. )
dt
Y
I
)
I
)
dt
y
I
)
First, consider the averaging scheme (3.106) . After some manipulation one ob -
tains
d dt = -t!..xt!.. " " [_ (U i+I,j - Ui-I,j + Vi ,j+1 - Vi,j -I)!/>?
Y
2t!..x
2t!..y
I.)
I
)
+ (Ui+l.j + Ui. j) !/>i+l,j!/>i,j _ (U i.j + Ui-I,j) !/>i.j!/>i-I ,j
+ (Vi.j+l/ Vi ,j)
_ (Vi,j +2 Vi ,j-1 )
J
I , )
2t!..x
+ v. . (!/>i'HI!/>i,j -!/>i,j-I!/>i. j)]
I .)
2t!..y
2
2
The first line on the right-hand side will vanish if the discrete velocity field satisfies (3.107). The terms in the second line sum to zero over the index i due to
the periodicity in x, and the terms in the third line sum to zero over the index j
due to the periodicity in y. Thus the averaging scheme (3.106) conserves 1I!/>1I2 in
problems where the flow field satisfies the discrete continuity equation (3.107).
Now consider the nonaveraging scheme (3.107) and again assurne that the problem is periodic in x and y . Then
d
= -2t!..xt!.. " " [u . . (!/>i+I,j!/>i . j -!/>i-I ,j!/>i,j)
= t!..xt!..y 4= [(
I
)
Ui,j
+ (
!/>i,j!/>i-I ,j
Vi,j - Vi,j-I)
t!..y
]
!/>i.j!/>i,j-I ,
where the last equality is obtained using the periodicity in x and y. It follows
that 111/>112 is not conserved by the nonaveraging scheme, regardless of the numerical form of the discrete continuity equation. In summary, (3.106) is preferable
to (3.105) because it better preserves the 12-norm stability and the conservation
3. Beyond the One-Way Wave Equation
yield a skew-symmetric matrix even when the velocities satisfy (3.107), and as a
consequence, it does not guarantee the conservation of lIull2.
As an alternative to the construction of A and the evaluation of its symmetry
properties, one can examine the stability and conservation properties of approximate solutions to (3.106) and (3.107) using the energy method to compute the
time tendency of the 12-norm of the serni-discrete solution from the relation
=!.. (",,!/>?t!..xt!.. ) =2""!/>. .d!/>i,j t!..xt!.. .
u't
dt
I ,)
Y
I. )
dt
Y
I
)
I
)
dt
y
I
)
First, consider the averaging scheme (3.106) . After some manipulation one ob -
tains
d dt = -t!..xt!.. " " [_ (U i+I,j - Ui-I,j + Vi ,j+1 - Vi,j -I)!/>?
Y
2t!..x
2t!..y
I.)
I
)
+ (Ui+l.j + Ui. j) !/>i+l,j!/>i,j _ (U i.j + Ui-I,j) !/>i.j!/>i-I ,j
+ (Vi.j+l/ Vi ,j)
_ (Vi,j +2 Vi ,j-1 )
J
I , )
2t!..x
+ v. . (!/>i'HI!/>i,j -!/>i,j-I!/>i. j)]
I .)
2t!..y
2
2
The first line on the right-hand side will vanish if the discrete velocity field satisfies (3.107). The terms in the second line sum to zero over the index i due to
the periodicity in x, and the terms in the third line sum to zero over the index j
due to the periodicity in y. Thus the averaging scheme (3.106) conserves 1I!/>1I2 in
problems where the flow field satisfies the discrete continuity equation (3.107).
Now consider the nonaveraging scheme (3.107) and again assurne that the problem is periodic in x and y . Then
d
= -2t!..xt!.. " " [u . . (!/>i+I,j!/>i . j -!/>i-I ,j!/>i,j)
= t!..xt!..y 4= [(
I
)
Ui,j
+ (
!/>i,j!/>i-I ,j
Vi,j - Vi,j-I)
t!..y
]
!/>i.j!/>i,j-I ,
where the last equality is obtained using the periodicity in x and y. It follows
that 111/>112 is not conserved by the nonaveraging scheme, regardless of the numerical form of the discrete continuity equation. In summary, (3.106) is preferable
to (3.105) because it better preserves the 12-norm stability and the conservation