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We describe a paradigm for multiscale modeling that combines the Mori-Zwanzig(MZ) formalism of Statistical Mechanics with the Variational Multiscale (VMS)method. The MZ-VMS approach leverages both VMS scale-separation projectors aswell as phase-space projectors to provide a systematic modeling approach thatis applicable to non-linear partial differential equations. Spectral as well ascontinuous and discontinuous finite element methods are considered. Theframework leads to a formally closed equation in which the effect of theunresolved scales on the resolved scales is non-local in time and appears as aconvolution or memory integral. The resulting non-Markovian system is used as astarting point for model development. We discover that unresolved scales leadto memory effects that are driven by an orthogonal projection of thecoarse-scale residual and inter-element jumps. It is further shown that anMZ-based finite memory model is a variant of the well-knownadjoint-stabilization method. For hyperbolic equations, this stabilization isshown to have the form of an artificial viscosity term. We further establishconnections between the memory kernel and approximate Riemann solvers. It isdemonstrated that, in the case of one-dimensional linear advection, theassumption of a finite memory and a linear quadrature leads to a closure termthat is formally equivalent to an upwind flux correction.

A Unified Framework for Multiscale Modeling using the Mori-Zwanzig Formalism and the Variational Multiscale Method