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The Filter Operation on Nonlinear Partial Differential Equations pp. 447-460 $100.00
Authors:  Garry Pantelis
It is often the case that computationally based solution methods are inadequate when
applied to systems of partial differential equations (PDE) which generate solutions which
contain nonlinear oscillations. The problem is one of resolution. The PDE systems of this
kind are often reformulated by introducing a filtering process which effectively removes the
frequencies of the unresolvable oscillations. This involves the procedure of applying some
kind of filter to the original system of PDE to derive a new system of governing equations
which aim to capture the macroscale behaviour of the solutions. The difficulty arises from
the presence of the nonlinear terms in the original system of PDE which result in nonlinear
filtered fields requiring some kind of closure.
Recently the author has proposed a procedure for the general construction of closure
models based on a consistency condition derived from the model error [1]. The approach
of [1] involves a transformation from equations governing the macroscale system to those
governing the microscale system. As a result only general structures of the macroscale
equations can be investigated with the necessity to incorporate into these equations a relatively
large number of unknown empirical parameters. Applying constraints particular to
the physics of each application can considerably reduce the number of empirical parameters
leading to expressions for the residuals which approaches practical application. The
same transformation technique applied to the inverse map, ie. from the microscale to the
macroscale systems, is less informative since relatively high frequency oscillations contained
in the microscale solutions are not removed and useful definitions of the model error are
more difficult to construct. As will be demonstrated here this difficulty can be overcome by
effectively incorporating into the inverse transformation the filtering process to remove the
oscillations of relatively high frequencies. The result is a formulation for the residuals which
satisfy the consistency condition defined in [1] but with much of the empiricism removed. 

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The Filter Operation on Nonlinear Partial Differential Equations pp. 447-460