332
11. Efficiency and Accuracy Improvement
the avoidable errors, which are due to inappropriate or improper use of the
code.
Many user errors are due to incorrect input data; often the error is found
only after many computations have been carried out - and sometimes it
is never found! Frequent errors are due to geometry scaling or parameter
selection, when dimensionless form of the equations is used. Another kind of
user error is due to a poor numerical grid (an inadequate distribution of grid
points can increase the errors by an order of magnitude or more - or prevent
one from getting solution at all).
11.1.2 Estimation of Errors
Every numerical solution contains errors; the important thing is t o know
how big the errors are, and whether their level is acceptable in the particular
application. The acceptable level of error can vary enormously. What may be
an acceptable error in an optimization study in the early design stage of a
new product, where only qualitative analysis and the response of the system
to design changes is important, could be catastrophic in another application.
It is thus as important to know how good the solution is for the particular
application as it is to obtain the solution in the first place. Especially when
using commercial codes, the user should concentrate on a careful analysis of
the results and on estimation of the errors, as far as possible. This may be
a great burden for a beginner, but an experienced CFD practitioner will do
this routinely.
Error analysis should be done in an order reversed from the order in
which they were introduced above. That is, one should begin by estimating
the iteration error (which can be done within a single calculation), then
the discretization error (which requires a minimum of two calculations on
different grids) and, finally, the modeling error (which may require many
calculations). Each of these should be an order of magnitude smaller than
the one it precedes or the estimation of the later errors will not be sufficiently
accurate.
Estimation of Iteration Errors. Knowing when to stop the iteration process is crucial from the point of view of computational efficiency. As a rule of
thumb, the iteration errors (sometimes also called convergence errors) should
be at least an order of magnitude lower than discretization errors. There is
no point in iterating to the round-off level; for most engineering applications,
relative accuracy (error compared to a reference value) of the three to four
significant digits in any variable is more than sufficient.
There are a number of ways of estimating these errors; Ferziger and PeriC
(1996) analyzed three of them in detail; see also Sect. 5.7. It can be shown that
the rate of reduction of error is the same as rate at which the residual and the
difference between successive iterates are reduced, except in the initial stage
of iteration. This was demonstrated in Fig. 7.10: the curves for the norm of
11. Efficiency and Accuracy Improvement
the avoidable errors, which are due to inappropriate or improper use of the
code.
Many user errors are due to incorrect input data; often the error is found
only after many computations have been carried out - and sometimes it
is never found! Frequent errors are due to geometry scaling or parameter
selection, when dimensionless form of the equations is used. Another kind of
user error is due to a poor numerical grid (an inadequate distribution of grid
points can increase the errors by an order of magnitude or more - or prevent
one from getting solution at all).
11.1.2 Estimation of Errors
Every numerical solution contains errors; the important thing is t o know
how big the errors are, and whether their level is acceptable in the particular
application. The acceptable level of error can vary enormously. What may be
an acceptable error in an optimization study in the early design stage of a
new product, where only qualitative analysis and the response of the system
to design changes is important, could be catastrophic in another application.
It is thus as important to know how good the solution is for the particular
application as it is to obtain the solution in the first place. Especially when
using commercial codes, the user should concentrate on a careful analysis of
the results and on estimation of the errors, as far as possible. This may be
a great burden for a beginner, but an experienced CFD practitioner will do
this routinely.
Error analysis should be done in an order reversed from the order in
which they were introduced above. That is, one should begin by estimating
the iteration error (which can be done within a single calculation), then
the discretization error (which requires a minimum of two calculations on
different grids) and, finally, the modeling error (which may require many
calculations). Each of these should be an order of magnitude smaller than
the one it precedes or the estimation of the later errors will not be sufficiently
accurate.
Estimation of Iteration Errors. Knowing when to stop the iteration process is crucial from the point of view of computational efficiency. As a rule of
thumb, the iteration errors (sometimes also called convergence errors) should
be at least an order of magnitude lower than discretization errors. There is
no point in iterating to the round-off level; for most engineering applications,
relative accuracy (error compared to a reference value) of the three to four
significant digits in any variable is more than sufficient.
There are a number of ways of estimating these errors; Ferziger and PeriC
(1996) analyzed three of them in detail; see also Sect. 5.7. It can be shown that
the rate of reduction of error is the same as rate at which the residual and the
difference between successive iterates are reduced, except in the initial stage
of iteration. This was demonstrated in Fig. 7.10: the curves for the norm of