Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

alternatives. From the Taylor expansion we have
and assuming ˆ 0:5
n j…x ÿ †
n @
ÿ n
@x ‡
2
2
n
2 n
@
@x2 ‡ O… t3 † …2:79†
1 @
2 @x
@
k
@x j…x ÿ †
1 @
2 @x
@
k
@x
@
ÿ
2 @x
@
@x
@
k
@x
‡ O… t 2 † …2:80a†
1
2 Qj…x ÿ † ˆ Qn
2 ÿ @Q
2
n
@x
…2:80b†
where is the distance travelled by the particle in the x-direction (Fig. 2.13) which is
n
ˆ U t …2:81†
where U is an average value of U along the characteristic. Di€erent approximations
of U lead to di€erent stabilizing terms. The following relation is commonly used 62;63
U ˆ U n ÿ U n
t @Un
@x
Inserting Eqs (2.79)±(2.82) into Eq. (2.78) we have
where
and
n ‡ 1 ÿ n ˆÿ t U @ n
‡ t
@
@x
t
2
k @
@x
@x
ÿ @
@x
@
@x U2 @
@x
n ‡ 1=2
k @
@x
ˆ 1 @
2 @x
ÿ t
2
n ‡ 1=2
k @
@x
n ‡ 1=2
‡ Q
@2 @
U k
@x2 @x
n ‡ 1
‡ 1 @
2 @x
Characteristic-based methods 39
t @Q
‡ U
2 @x
k @
@x
n
n
…2:82†
…2:83a†
…2:83b†
Q n ‡ 1=2 ˆ Qn ‡ 1 ‡ Q n
…2:83c†
2
In the above equation, higher-order terms (from Eq. 2.80) are neglected. This, as
already mentioned, is of an identical form to that resulting from Taylor±Galerkin
procedures which will be discussed fully in the next section, and the additional
terms add the stabilizing di€usion in the streamline direction. For multidimensional
problems, Eq. (2.83a) can be written in indicial notation and approximating n ‡ 1=2
terms with n terms (for the fully explicit form)
n ‡ 1 n @
ÿ ˆÿ t Uj ÿ
@xj @
k
@xi @
n
‡ Q
@xi t @ @
‡ t U
2 @x
iUj ÿ
i @xj t
2 U @ @
k k
@xk @xi @
‡
@xi t
2 U n
@Q
i
@xi …2:84†