124
5. Solution of Linear Equation Systems
additional effort is much smaller than that needed to treat the entire higherorder approximation implicitly. One can also multiply the "old" term with
a blending factor between zero and unity to produce a mixture of the pure
low-order and pure high-order schemes. This is sometimes used to avoid oscillations which result when higher-order schemes are used on grids that are
not sufficiently fine. For example, when flow around a body is computed, one
would like to use a fine grid near the body and a coarser grid far from it.
A high-order scheme may produce oscillations in the coarse-grid region, thus
spoiling the whole solution. Since the variables vary slowly in the coarse-grid
region, we may reduce the order of approximation there without affecting
the solution in the fine-grid region; this can be achieved by using a blending
factor in the outer region only.
More details on other uses of deferred-correction approach will be given
in subsequent chapters.
5.7 Convergence Criteria and Iteration Errors
When using iterative solvers, it is important to know when to quit. The most
common procedure is based on the difference between two successive iterates;
the procedure is stopped when this difference, measured by some norm, is less
than a pre-selected value. Unfortunately, this difference may be small when
the error is not small and a proper normalization is essential.
From the the analysis presented in Sect. 5.3.2, we find (see Eqs. (5.14)
and (5.27)):
Bn = qY+' - 4" x (A1 - l)(Al)nalqbl ,
(5.83)
where bn is the difference between solution at iterations n + 1 and n , and A1
is the largest eigenvalue or spectral radius of the iteration matrix. It can be
estimated from:
where l l a l l represents the norm (e.g. root mean square or Lz norm) of a.
Once an estimate of the eigenvalue is available, it is not difficult to estimate the iteration error. In fact, by rearranging Eq. (5.83), we find:
A good estimate of the iteration error is therefore:
5. Solution of Linear Equation Systems
additional effort is much smaller than that needed to treat the entire higherorder approximation implicitly. One can also multiply the "old" term with
a blending factor between zero and unity to produce a mixture of the pure
low-order and pure high-order schemes. This is sometimes used to avoid oscillations which result when higher-order schemes are used on grids that are
not sufficiently fine. For example, when flow around a body is computed, one
would like to use a fine grid near the body and a coarser grid far from it.
A high-order scheme may produce oscillations in the coarse-grid region, thus
spoiling the whole solution. Since the variables vary slowly in the coarse-grid
region, we may reduce the order of approximation there without affecting
the solution in the fine-grid region; this can be achieved by using a blending
factor in the outer region only.
More details on other uses of deferred-correction approach will be given
in subsequent chapters.
5.7 Convergence Criteria and Iteration Errors
When using iterative solvers, it is important to know when to quit. The most
common procedure is based on the difference between two successive iterates;
the procedure is stopped when this difference, measured by some norm, is less
than a pre-selected value. Unfortunately, this difference may be small when
the error is not small and a proper normalization is essential.
From the the analysis presented in Sect. 5.3.2, we find (see Eqs. (5.14)
and (5.27)):
Bn = qY+' - 4" x (A1 - l)(Al)nalqbl ,
(5.83)
where bn is the difference between solution at iterations n + 1 and n , and A1
is the largest eigenvalue or spectral radius of the iteration matrix. It can be
estimated from:
where l l a l l represents the norm (e.g. root mean square or Lz norm) of a.
Once an estimate of the eigenvalue is available, it is not difficult to estimate the iteration error. In fact, by rearranging Eq. (5.83), we find:
A good estimate of the iteration error is therefore: