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Development of an error-minimizing adaptive grid method
M Borsboom
Delft Hydraulics, P.O. box 177,
2600 MH Delft, The Netherlands.
E-mail: { tt mart.borsboom@wldelft.nl
Abstract
In our search for a suitable approximation of the numerical simulation
error, to be used as the monitor function in a moving adaptive grid technique,
we found that there is often a discrepancy between the discretization of
the physical model under study (flow and transport equations in our case)
and the way the numerical solution is represented.
For example, the use of piecewise linear polynomials to reconstruct the
numerical solution over the whole domain is consistent with the use of a
second-order accurate discretization technique, but this does not imply that
the second-order interpolation error is representative of the numerical
discretization error.
We have solved this problem by not designing an adaptive grid technique for
a particular discretization method, but by developing a discretization
method that permits a consistent error analysis.
Using this method, the leading truncation error of discretized flow and
transport equations can be reformulated into numerical simulation errors
that are of the same form and size as the leading interpolation errors
of the numerical function approximations used in the construction of these
discretizations.
To our knowledge, there exists no other scheme which has this `compatibility'
property.
The numerical error is measured in the 
-norm, to assure that it is
O(
) in the neighborhood of discontinuities which is consistent
with the local order of accuracy of any discretization method.
Measuring the error in 
-norm on the other hand would indicate
an O(1) error near discontinuities, independent of the size of the grid,
making error minimization through grid adaptation an indeterminate problem.
Under the assumption that the numerical solution is sufficiently smooth, it
can be shown, at least in 1D, that error minimization is approximately
equivalent with the equidistribution of the error per grid cell.
In order to develop an iterative solution method suitable for the highly
nonlinear equidistribution problem, we have first concentrated on a simple
test problem.
The second-order accurate compatible scheme was used in combination with
the error-minimizing grid adaptation strategy to determine as accurately as
possible the numerical approximation 
of a given function.
Upon convergence, the grid points are placed in such a way that the
leading error term per grid cell, 
, is
equidistributed.
Higher-order artificial smoothing has been introduced to assure that all
numerical variables are sufficiently smooth.
Applying the adaptive grid technique to piecewise constant functions turned
out to be not straightforward.
Because of the -norm error measure, grid points are distributed
automatically and evenly over all specified discontinuities, but tend to
form very dense clusters.
Recently obtained results are however quite satisfactory, e.g.,
approximating a function with two discontinuities using 50 grid cells
resulted in local adaptive grid refinements of more than a factor 1000.
This result was obtained in about 10 grid iteration steps.
Approximating (partly) smooth functions on the other hand is much easier
and converges much faster, since then the maximum possible local grid
refinements are limited.
Some work has been done to apply the method to the 1D shallow-water
equations, for the accurate modeling of bores (moving wave fronts)
and hydraulic jumps.
Last modified Fri Jun 21 19:19:11 GB-Eire 1996
(DBD)