334
11. Efficiency and Accuracy Improvement
the problem by reducing the Reynolds number or computing the flow for one
half of the geometry, using a symmetry boundary condition - or perform a
transient computation.
Estimation of Discretization Errors. Discretization errors can only be
estimated if solutions on systematically refined grids are compared; see Sect.
3.10.2 and 3.9 for more details. As noted earlier, these errors are due to
the use of approximations for the various terms in the equations and the
boundary conditions. For problems with smooth solutions, the quality of an
approximation is described in terms of its order, which relates the truncation
error of the approximation to the grid spacing to a some power; if the truncation error of a spatial derivative is proportional to say (Az)P, we say that
the approximation is of pth order. The order is not a direct measure of the
magnitude of the error; it indicates how the error changes when the spacing is
changed. Approximations of the same order may have errors which differ as
much as an order of magnitude; also, an approximation of a lower order may
have a smaller error for a particular grid than one of higher order. However,
as the spacing becomes smaller, the higher-order approximation will certainly
become more accurate.
It is easy to find the order of many approximations using Taylor series
expansion. On the other hand, different approximations may be used for
different terms, so the order of the solution method as a whole may not be
obvious (it is usually of the order of the least accurate approximation of a
significant term in the equation). Also, errors in the implementation of the
algorithm in the computer code may yield a different order than expected.
It is therefore important to check the order of the method for each class of
problems using the actual code.
The best way to analyze discretization errors on structured grids is to
halve the spacing in each direction. However, this is not always possible; in
3D, this requires an eight-fold increase in the number of nodes. Thus, the third
grid has 64 times as many points, and we may not be able to afford another
refinement level. On the other hand, the errors are usually not uniformly
distributed, so there is no point in refining the whole grid. Furthermore, when
unstructured grids with arbitrary control volumes or elements are used, there
are no local coordinate directions and the elements are refined in a different
manner.
What is important is that the refinement is substantial and systematic.
Increasing the number of nodes in one direction from say 54 to 62 is not
very useful, except in an academic problem with uniform error distribution
and a uniform grid; the refined grid should have at least 50 % more nodes
in each direction than the original grid. Systematic refinement means that
the grid topology and relative spatial density of grid points should remain
comparable on all grid levels. A different distribution of grid points may lead
to substantial changes in discretization errors without changing the number
of nodes. An example is shown in Fig. 7.11: the results obtained on a non-
11. Efficiency and Accuracy Improvement
the problem by reducing the Reynolds number or computing the flow for one
half of the geometry, using a symmetry boundary condition - or perform a
transient computation.
Estimation of Discretization Errors. Discretization errors can only be
estimated if solutions on systematically refined grids are compared; see Sect.
3.10.2 and 3.9 for more details. As noted earlier, these errors are due to
the use of approximations for the various terms in the equations and the
boundary conditions. For problems with smooth solutions, the quality of an
approximation is described in terms of its order, which relates the truncation
error of the approximation to the grid spacing to a some power; if the truncation error of a spatial derivative is proportional to say (Az)P, we say that
the approximation is of pth order. The order is not a direct measure of the
magnitude of the error; it indicates how the error changes when the spacing is
changed. Approximations of the same order may have errors which differ as
much as an order of magnitude; also, an approximation of a lower order may
have a smaller error for a particular grid than one of higher order. However,
as the spacing becomes smaller, the higher-order approximation will certainly
become more accurate.
It is easy to find the order of many approximations using Taylor series
expansion. On the other hand, different approximations may be used for
different terms, so the order of the solution method as a whole may not be
obvious (it is usually of the order of the least accurate approximation of a
significant term in the equation). Also, errors in the implementation of the
algorithm in the computer code may yield a different order than expected.
It is therefore important to check the order of the method for each class of
problems using the actual code.
The best way to analyze discretization errors on structured grids is to
halve the spacing in each direction. However, this is not always possible; in
3D, this requires an eight-fold increase in the number of nodes. Thus, the third
grid has 64 times as many points, and we may not be able to afford another
refinement level. On the other hand, the errors are usually not uniformly
distributed, so there is no point in refining the whole grid. Furthermore, when
unstructured grids with arbitrary control volumes or elements are used, there
are no local coordinate directions and the elements are refined in a different
manner.
What is important is that the refinement is substantial and systematic.
Increasing the number of nodes in one direction from say 54 to 62 is not
very useful, except in an academic problem with uniform error distribution
and a uniform grid; the refined grid should have at least 50 % more nodes
in each direction than the original grid. Systematic refinement means that
the grid topology and relative spatial density of grid points should remain
comparable on all grid levels. A different distribution of grid points may lead
to substantial changes in discretization errors without changing the number
of nodes. An example is shown in Fig. 7.11: the results obtained on a non-