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

2.7 Taylor±Galerkin procedures for scalar variables
In the Taylor±Galerkin process, the Taylor expansion in time precedes the Galerkin
space discretization. Firstly, the scalar variable is expanded by the Taylor series in
time 58;66
n ‡ 1 ˆ n ‡ t @ n
From Eq. (2.61a) we have
@ n
@t
and
2 n
@ @
ˆ 2 @t @t
@t
‡ t2
2
@ @
ˆ ÿU ‡
@x @x
@ @
ÿU ‡
@x @x
2 n
@
@t2 ‡ O… t3 † …2:108†
k @
@x
k @
@x
‡ Q
n
‡ Q
Substituting Eqs (2.109) and (2.110) into Eq. (2.108) we have
n ‡ 1 n @ @
ÿ ˆÿ t U ÿ
@x @x
k @
@x
‡ Q
Assuming U and k to be constant we have
n ‡ 1 n @ @
ÿ ˆÿ t U ÿ
@x @x
k @
@x
‡ Q
n
n
ÿ t2
2
ÿ t2
2
@
@t
@
@x
n
@ @
U ÿ
@x @x
@ @
U ÿ
@t @x
Inserting Eq. (2.109) into Eq. (2.112) and neglecting higher-order terms
n ‡ 1 n @ @
ÿ ˆÿ t U ÿ
@x @x
‡ t2
2
k @
@x
@
@x U2 @ @
ÿ U
@x @x
‡ Q
n
k @
@x
‡ UQ
n
k @
@x
k @
@t
…2:109†
…2:110†
‡ Q
n
…2:111†
‡ Q
n
…2:112†
‡ O… t 3 † …2:113†
As we can see the above equation, having assumed constant U and k, is identical to
Eq. (2.83a) derived from the characteristic approach. Clearly for scalar variables both
characteristic and Taylor±Galerkin procedures give identical stabilizing terms. Thus
selection of a method for a scalar variable is a matter of taste. However, the sound
mathematical justi®cation of the characteristic±Galerkin method should be
mentioned here.
The Taylor±Galerkin procedure for the convection±di€usion equation in multidimensions
can be written as
n ‡ 1 n
ÿ ˆÿ t Uj
@xj ÿ t
2
@
@x i
@
ÿ @
@x i
@
UiUj @x j
Taylor±Galerkin procedures for scalar variables 47
k @
@xi @
ÿ U i
@x j
‡ Q
k @
@x j
‡ U iQ …2:114†