Biomedical Engineering Reference
In-Depth Information
a specified tolerance at every nodal location. Secondly, the numerical solution no
longer changes with additional iterations. Thirdly, overall mass, momentum, energy
and scalar balances are obtained. During the numerical procedure, the imbalances
(errors) of the discretised equations are monitored and these defects are commonly
referred to as the residuals of the system of algebraic equations, i.e. they measure
the extent of imbalances arising from these equations and terminate the numerical
process when a specified tolerance is reached. For satisfactory convergence, the
residuals
should diminish as the numerical process progresses. In the likelihood that
the imbalances grow, as reflected by increasing residual values, the numerical solu-
tion is thus classified as being divergent. It is noted that iterative convergence is not
the same as grid convergence. Grid convergence seeks a grid independent solution.
The previous discussion of
convergence
,
consistency
and
stability
has been pri-
marily concerned with the solution behavior where the finite quantities, such as the
time step
t
and mesh spacing
x
,
y
and
z
, diminish. Since the discretised forms
of the transport equations governing the flow and energy transfer are always solved
numerically on a finite grid layout, the solution obtained is always approximate. The
corresponding issue of
accuracy
therefore becomes another important consideration.
One method where accuracy can be assessed on a finite grid is to apply it to a
related but simplified problem that possesses an exact solution. However, accuracy
is usually problem dependent; an algorithm that is accurate for one model problem
may not necessarily be as accurate as for another more complicated problem. Another
probable way for assessing accuracy is to obtain solutions on successively refined
grids (mesh indpendence) and to check that, with successive refinements, the solution
is not changing satisfying some predetermined accuracy. This technique assumes that
the approximate solutions will converge to the exact solution as the finite quantities
diminish and then the approximate solution on the finest grid can be used in place
of the exact solution. Assuming that the accuracy of this approximate solution can
be assessed it is important to consider the related question of how accuracy may
be improved. It is important the reader be aware that
a converged solution does
not necessarily mean an accurate solution
. Some possible sources of solution error
can occur through numerical calculations of the algebraic equations. If these errors
are to be minimized, some systematic steps to perform numerical analysis can be
performed.
In summary, the conceptual framework linking the various aspects of
consistency
,
stability
,
convergence
and
accuracy
beginning from the governing partial differential
equations as considered and arriving at the approximate solution of the algebraic
equations can be seen in Fig.
7.29
.
For more indepth material on these concepts, the interested reader is referred
to the following textbooks and journal publications for the following: consistency
(Ferziger and Peric 1999; Fletcher 1991); stability (Tu et al. 2008; Yeoh and Tu 2009);
convergence (Roy 2003; Tannehill et al. 1997); accuracy (AIAA 1998; Wendt and
Anderson 2009).
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