Geoscience Reference
In-Depth Information
Figure 8.2
Interface arrangements for multiblock grid: (a) patched, (b) simple overlapping, and
(c) complex overlapping.
patched grids, two neighboring blocks are connected at a common grid line without
overlapping, while on the overlapping grids, the blocks can be superimposed arbitrarily
on each other to cover the domain of interest. Compared to the patched grids, the
overlapping grids have more flexibility in grid generation but may be less accurate due
to interpolation errors as information is exchanged between blocks.
The discretization of the governing equations in each block is the same as in a single-
block domain. However, for a complete solution, internal boundary conditions should
be applied at the interfaces. To avoid errors and the generation of spurious numerical
oscillations, the conservation laws should be satisfied at the interfaces. During the
solution process, the information updated at each time or iteration step needs to be
transferred between the blocks. Therefore, the key issues that affect the performance
of the multiblock method are interface treatment and information exchange between
blocks.
8.1.2 Multiblock method for 1-D problems
Let us consider the 1-D steady convection-diffusion problem (4.15) as an example of
the multiblock method (Shyy
et al
., 1997). For the sake of simplicity, only two grid
blocks are used. The grids in these two blocks are shown in Fig. 8.3. Let
φ
l
,
i
denote the
discrete approximation to
φ
at the point
i
(
=
1, 2,
...
,
N
k
) of the component grid block
l
1, 2). Eq. (4.15) can be discretized using the generic numerical scheme introduced
in Chapter 4 as
(
=
a
Pl
,
i
φ
=
a
Wl
,
i
φ
+
a
El
,
i
φ
(8.1)
l
,
i
l
,
i
−
1
l
,
i
+
1
where
a
Pl
,
i
,
a
El
,
i
, and
a
Wl
,
i
are coefficients.
The discretized equations on each block can be solved with a direct or iterative
method. However, boundary conditions have to be provided at the interface to connect
the solutions on the two blocks so that a global solution can be obtained. In the solution
