  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 as
well as phase-space projectors to provide a systematic modeling approach that
is applicable to non-linear partial differential equations. Spectral as well as
continuous and discontinuous finite element methods are considered. The
framework leads to a formally closed equation in which the effect of the
unresolved scales on the resolved scales is non-local in time and appears as a
convolution or memory integral. The resulting non-Markovian system is used as a
starting point for model development. We discover that unresolved scales lead
to memory effects that are driven by an orthogonal projection of the
coarse-scale residual and inter-element jumps. It is further shown that an
MZ-based finite memory model is a variant of the well-known
adjoint-stabilization method. For hyperbolic equations, this stabilization is
shown to have the form of an artificial viscosity term. We further establish
connections between the memory kernel and approximate Riemann solvers. It is
demonstrated that, in the case of one-dimensional linear advection, the
assumption of a finite memory and a linear quadrature leads to a closure term
that is formally equivalent to an upwind flux correction.

Numerical Analysis