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

52 Convection dominated problems
C = 0.1, C Lap = 0
C = 0.1, C Lap = 1
Fig. 2.21 Propagation of a steep front in Burger's equation with solution obtained using different values of
Lapidus C v ˆ C Lap.
Part III: Vector-valued functions
2.10 Vector-valued variables
2.10.1 The Taylor±Galerkin method used for vector-valued
variables
The only method which adapts itself easily to the treatment of vector variables is that
of the Taylor±Galerkin procedure. Here we can repeat the steps of Sec. 2.8 but now
addressed to the vector-valued equation with which we started this chapter (Eq. 2.1).
Noting that now has multiple components, expanding by a Taylor series in time
we have 66;70
n ‡ 1 n @
ˆ ‡ t ‡
@t n
t2 @
2
2
@t2 where is a number such that 0 4 4 1.
From Eq. (2.1),
@
ˆÿ
@t n
@Fi ‡
@xi @Gi ‡ Q
@xi n
n ‡
…2:124†
…2:125a†