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

38 Convection dominated problems
accuracy of such integration. 46 The scheme is stable and indeed exact as far as the
convective terms are concerned if the integration is performed exactly (which of
course is an unreachable goal). However, stability and indeed accuracy will even
then be controlled by the di€usion terms where several approximations have been
involved.
2.6.3 A simple explicit characteristic±Galerkin procedure
Many variants of the schemes described in the previous section are possible and were
introduced quite early. References 45±56 present some successful versions. However,
all methods then proposed are somewhat complex in programming and are time consuming.
For this reason a simpler alternative was developed in which the di culties
are avoided at the expense of conditional stability. This method was ®rst published in
1984 57 and is fully described in numerous publications. 58ÿ61 Its derivation involves a
local Taylor expansion and we illustrate this in Fig. 2.13.
We can write Eq. (2.61a) along the characteristic as
@
@t …x0…t†; t†ÿ @ @
0 k
@x @x0 ÿ Q…x0 †ˆ0 …2:77†
As we can see, in the moving coordinate x 0 , the convective acceleration term
disappears and source and di€usion terms are averaged quantities along the characteristic.
Now the equation is self-adjoint and the Galerkin spatial approximation is
optimal. The time discretization of the above equation along the characteristic
(Fig. 2.13) gives
1
t … n ‡ 1 ÿ n j …x ÿ ††
@
@x
k @
@x
‡…1 ÿ † @
@x
ÿ Q
k @
@x
n ‡ 1
ÿ Q
n
j …x ÿ †
…2:78†
where is equal to zero for explicit forms and between zero and unity for semi- and
fully implicit forms. As we know, the solution of the above equation in moving
coordinates leads to mesh updating and presents di culties, so we will suggest
t
φ n (x – δ)
Fig. 2.13. A simple characteristic±Galerkin procedure.
x
δ≈U∆t
n + 1 φ
φ n