The Filter Operation on Nonlinear Partial Differential Equations pp. 447-460
Authors: Garry Pantelis
Abstract: 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 . The approach of  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  but with much of the empiricism removed.