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weighted Sobolev norm, Kirby optimized the approximation of the higher derivative terms in numerical

simulations of PDE, and reduced the size of the associated systems of ODE. The goal of Kirby's

(1992) extension is to minimize (Wo[U uNI 2 ~- ]4)IIux 2 + w21Uxx

+ - - . > , which is a kind

of global inner product optimal condition.

It should be noted that the POD method or its extensions cannot be used for the situations in which

the optimal conditions are other than the inner product global ones. Sometimes the most essential

characters of the system are unable to be presented by the optimal condition of global inner product,

and with such optimal conditions it is impossible to focus one's attention on some key region, such

as the production of vorticity on the boundaries, or to accurately describe some important flow

phenomena in a certain time period, such as the breakdown of vortices.

The work presented here puts forward an optimal theory for an expansion of flow quantities

to captur

simulations of PDE, and reduced the size of the associated systems of ODE. The goal of Kirby's

(1992) extension is to minimize (Wo[U uNI 2 ~- ]4)IIux 2 + w21Uxx

+ - - . > , which is a kind

of global inner product optimal condition.

It should be noted that the POD method or its extensions cannot be used for the situations in which

the optimal conditions are other than the inner product global ones. Sometimes the most essential

characters of the system are unable to be presented by the optimal condition of global inner product,

and with such optimal conditions it is impossible to focus one's attention on some key region, such

as the production of vorticity on the boundaries, or to accurately describe some important flow

phenomena in a certain time period, such as the breakdown of vortices.

The work presented here puts forward an optimal theory for an expansion of flow quantities

to captur

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