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

2.6 Characteristic-based methods
2.6.1 Mesh updating and interpolation methods
We have already observed that, if the spatial coordinate is `convected' in the manner
implied by Eq. (2.62), i.e. along the problem characteristics, then the convective, ®rstorder,
terms disappear and the remaining problem is that of simple di€usion for
which standard discretization procedures with the Galerkin spatial approximation
are optimal (in the energy norm sense).
The most obvious use of this in the ®nite element context is to update the position
of the mesh points in a lagrangian manner. In Fig. 2.11(a) we show such an update for
the one-dimensional problem of Eq. (2.61) occurring in an interval t.
For a constant x 0 coordinate
dx ˆ U dt …2:66†
(a) Forward
t n
t n–1 = t n – ∆t
(b) Backward
t
t
t n+1 = t n + ∆t
t n
∆t
∆t
Fig. 2.11 Mesh updating and interpolation: (a) Forward; (b) Backward.
h
Initial node
position
Characteristic-based methods 35
Characteristic
Updated node
position
x
x