  The development of reduced models for complex multiscale problems remains one
of the principal challenges in computational physics. The optimal prediction
framework of Chorin et al., which is a reformulation of the Mori-Zwanzig (M-Z)
formalism of non-equilibrium statistical mechanics, provides a methodology for
the development of mathematically-derived reduced models of dynamical systems.
Several promising models have emerged from the optimal prediction community and
have found application in molecular dynamics and turbulent flows. In this work,
a new M-Z-based closure model that addresses some of the deficiencies of
existing methods is developed. The model is constructed by exploiting
similarities between two levels of coarse-graining via the Germano identity of
fluid mechanics and by assuming that memory effects have a finite temporal
support. The appeal of the proposed model, which will be referred to as the
`dynamic-MZ-$\tau$' model, is that it is parameter-free and has a structural
form imposed by the mathematics of the coarse-graining process (rather than the
phenomenological assumptions made by the modeler, such as in classical subgrid
scale models). To promote the applicability of M-Z models in general, two
procedures are presented to compute the resulting model form, helping to bypass
the tedious error-prone algebra that has proven to be a hindrance to the
construction of M-Z-based models for complex dynamical systems. While the new
formulation is applicable to the solution of general partial differential
equations, demonstrations are presented in the context of Large Eddy Simulation
closures for Burgers equation, decaying homogeneous turbulence, and turbulent
channel flow. The performance of the model and validity of the underlying
assumptions are investigated in detail.
