Geoscience Reference
In-Depth Information
(3) Solve the pressure-correction equation to obtain p , for blocks 1 to nb ;
(4) Calculate p n + 1 , U n + 1
x , and U n + y , for blocks 1 to nb ;
(5) Treat the corrected pressure p n + 1 as a new guessed pressure p , and repeat the
procedure from step 2 to step 6 until a converged solution is obtained;
(6) Conduct the calculation of sediment transport and bed change, if needed, for
blocks 1 to nb .
In principle, both strategies can be successful. Which one performs better is problem-
dependent. Usually, the first strategy needs less memory than the second strategy,
because the second strategy needs to store the coefficients for all blocks while the first
strategy needs to store the coefficients for only one block.
8.1.4 Efficiency of multiblock method
For a complex problem, the multiblock method requires less computer memory and
has more flexibility for grid generation than the single-block method. The task of
computations in each block can be assigned to a processor and, thus, the multiblock
method can be run on a parallel computer. However, the computations on all blocks
should be synchronized, and the information transferred between processors may not
be the “latest”; thus, some multiblock algorithms designed for serial (single-processor)
computers may not run efficiently on parallel computers. The efficiency depends on
the problem, the method of solving it, and the computer used.
The explicit schemes for an unsteady problem can be easily extended from single
block to multiblock algorithms. Because all operations are performed on data from
the previous time step, all that is needed is to transfer the data at the interface between
neighboring blocks after the completion of each time step. The sequences of operation
are identical when using single block andmultiblockmethods on either serial or parallel
computers; so are the results. For solving the algebraic equations that result from a
steady problem or from implicit schemes for an unsteady problem, the Jacobi iteration
method at each iteration step only needs the “old” values of the previous iteration
step, which is very similar to the manipulation of explicit schemes in the sense of time
step. Therefore, the explicit schemes and the Jacobi iteration method are very easy to
parallelize, with very efficient speed-up.
On serial computers, the multiblock Gauss-Seidel iteration method has the same
performance as the single-block version, if the sweeping order among blocks is in such
a sequence that the “latest” values of the variable in one block can be used for the
calculation in the adjacent block. For example, if the sweeping order for the point-to-
point iteration in each block is from the lower left point to the lower right point and
thence to the upper left point, the sweeping order among blocks must be also from the
lower left block to the lower right block and thence to the upper left block. However,
the multiblock Gauss-Seidel iteration method cannot be extended straightforwardly
from serial computers to parallel computers, because the “latest” values at the interface
between blocks are not available during the synchronized computations on all the
assigned processors. To solve this problem, a specially designed “red-black” ordering
method is often used (see Golub and Ortega, 1993). On a structured grid, the points
are imaged to be “colored” in the same way as a checkerboard, as shown in Fig. 8.7.
Search WWH ::




Custom Search