Page 281 - Introduction to Computational Fluid Dynamics

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CONVERGENCE ENHANCEMENT
the machine accuracy will permit but typically CC = 10 −5 (say) suffices for most
engineering applications.
The convergence rate C R may be defined as
dR
CR =− , (9.4)
dl
where l is the iteration level. Economic computations will require that CR must
be as high as possible. Algebraic equation solvers such as the GS, the TDMA, and
the ADI introduced in Chapter 5, however, demonstrate the following convergence
rate properties:

1. Overall CR is higher when A k and S are constants rather than when they are
dependent on .
2. The initial (small l) CR is high but progressively decreases as convergence is
approached.
3. CR is higher when the A k are small (for example, coarse grids) than when they
are large (fine grids).
4. CR is higher when Dirichlet boundary conditions are specified at all boundaries
than when Neumann (or gradient) boundary conditions are specified. This is one
reason why the pressure-correction equation is slow to converge.

5. The convergence history (i.e., R ∼ l relationship) is typically monotonic when
A k and S are constants but can be highly nonmonotonic (or oscillatory) when
the equations are strongly coupled.

This last point is concerned with the stability of the iterative procedure. The
reader may wish to relate this phenomenon with damping of waves discussed in
Chapter 3.
The CR of the basic iterative methods (GS and ADI for 2D problems) can be
enhanced by several techniques. Here, a few of them that have the facility of being
incorporated in a generalised computer code will be considered. It is important to
note, however, that all convergence enhancement techniques essentially take ever
greater account of the implicitness embodied in the equation set (9.1). Thus, it is
recognised that P is implicitly related not only to its immediate neighbours but also
to its distant neighbours. The objective, therefore, is to strengthen this relationship
with the distant neighbours.
The merit of this observation has already been sensed in Chapter 2, where
convergence rates of GS (point-by-point) and TDMA (line-by-line) procedures
were compared for a 1D problem. In this chapter, the main interest is to consider
2D problems. The enhancement techniques considered can also be extended to 3D
problems.