11.1 Error Analysis and Estimation
333
the residual, the norm of difference between successive iterates, the estimated
error, and the actual iteration error are all parallel after some iterations.
Therefore, if one knows the error level a t the start of computation (which
is the solution itself if one starts with zero fields and somewhat lower if a
rough but reasonable guess is made), then one can be confident that the error
will fall 2-3 orders of magnitude if the norm of residuals (or of differences
between two iterates) has fallen 3-4 orders of magnitude. This would mean
that the first two or three most significant digits will not change in further
iterations, and thus that the solution is accurate within 0.01-0.1 %.
A common error is to look a t the magnitude of the differences between
successive iterates and stop computations when they do not differ by more
than a certain small number. However, the difference can be small because
the iterations are slowly converging while the iteration error may be enormous. In order to estimate the magnitude of the error, one has to properly
normalize the difference between successive iterates; when the convergence
is slow, the normalization factor becomes large (see Sect. 5.7). On the other
hand, requiring that the norm of differences fall three to four orders of magnitude is usually a safe criterion. Since the linear equation solvers in most
CFD methods require the computation of residuals, the simplest practice is
to monitor their norm (the sum of absolute values or square root of the sum
of squares).
On a coarse grid, where the discretization errors are large, one can allow
larger iteration errors; tighter tolerance is required for fine grids. This is
automatically taken into account if the convergence criterion is based on the
sum of residuals and not on the average residual per node, since the sum grows
with growing number of nodes and thus tightens the convergence criterion.
When a new code is developed, or a new feature is added, one has to
demonstrate beyond reasonable doubt that the solution process does converge until the residuals reach the round-off level. Very often, the lack of such
convergence indicates that errors are present, especially in the implementation of boundary conditions. Sometimes, the limit is below the threshold at
which the convergence is declared and may not be noticed. In other cases, the
procedure may stop converging (or even diverge) much earlier. Once all new
features have been thoroughly tested, one can return to the usual convergence
criteria.
Also, if one tries t o obtain a steady solution for a problem which is inherently unsteady (e.g. flow around a circular cylinder at a Reynolds number
for which the von Karman vortex street is present), some iterative methods
may not converge. Since each iteration can be interpreted as a pseudo-time
step, it is likely that the process will not diverge, but that the residuals oscillate indefinitely. This often happens if the geometry is symmetric and the
steady symmetric solution is unstable (e.g. diffusers or sudden expansions;
steady solutions - both laminar and Reynolds-averaged - are asymmetric,
with a larger separation region on one side). One can check whether this is
333
the residual, the norm of difference between successive iterates, the estimated
error, and the actual iteration error are all parallel after some iterations.
Therefore, if one knows the error level a t the start of computation (which
is the solution itself if one starts with zero fields and somewhat lower if a
rough but reasonable guess is made), then one can be confident that the error
will fall 2-3 orders of magnitude if the norm of residuals (or of differences
between two iterates) has fallen 3-4 orders of magnitude. This would mean
that the first two or three most significant digits will not change in further
iterations, and thus that the solution is accurate within 0.01-0.1 %.
A common error is to look a t the magnitude of the differences between
successive iterates and stop computations when they do not differ by more
than a certain small number. However, the difference can be small because
the iterations are slowly converging while the iteration error may be enormous. In order to estimate the magnitude of the error, one has to properly
normalize the difference between successive iterates; when the convergence
is slow, the normalization factor becomes large (see Sect. 5.7). On the other
hand, requiring that the norm of differences fall three to four orders of magnitude is usually a safe criterion. Since the linear equation solvers in most
CFD methods require the computation of residuals, the simplest practice is
to monitor their norm (the sum of absolute values or square root of the sum
of squares).
On a coarse grid, where the discretization errors are large, one can allow
larger iteration errors; tighter tolerance is required for fine grids. This is
automatically taken into account if the convergence criterion is based on the
sum of residuals and not on the average residual per node, since the sum grows
with growing number of nodes and thus tightens the convergence criterion.
When a new code is developed, or a new feature is added, one has to
demonstrate beyond reasonable doubt that the solution process does converge until the residuals reach the round-off level. Very often, the lack of such
convergence indicates that errors are present, especially in the implementation of boundary conditions. Sometimes, the limit is below the threshold at
which the convergence is declared and may not be noticed. In other cases, the
procedure may stop converging (or even diverge) much earlier. Once all new
features have been thoroughly tested, one can return to the usual convergence
criteria.
Also, if one tries t o obtain a steady solution for a problem which is inherently unsteady (e.g. flow around a circular cylinder at a Reynolds number
for which the von Karman vortex street is present), some iterative methods
may not converge. Since each iteration can be interpreted as a pseudo-time
step, it is likely that the process will not diverge, but that the residuals oscillate indefinitely. This often happens if the geometry is symmetric and the
steady symmetric solution is unstable (e.g. diffusers or sudden expansions;
steady solutions - both laminar and Reynolds-averaged - are asymmetric,
with a larger separation region on one side). One can check whether this is