Geometric foundation of invariant and conservative parameterization schemes

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I propose a novel framework for developing physical parameterization schemes. The key paradigm is to use the geometric properties of the original governing equations of hydro-thermodynamics to formulate parameterization schemes that preserve these properties in the approximated equations. This will then form the basis for numerical models. These properties will include both symmetries and conservation laws, which are among the most important features of any physical model. The construction of geometry-preserving parameterization schemes will mainly rely on methods from the group analysis of differential equations.