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

TL;DR

This paper introduces a unified multiscale modeling framework combining the Mori-Zwanzig formalism with the Variational Multiscale method, applicable to nonlinear PDEs, capturing unresolved scale effects as non-local memory terms.

Contribution

It develops a systematic approach integrating MZ and VMS, revealing memory effects and connecting to existing stabilization and flux correction methods.

Findings

Memory effects driven by unresolved scales are characterized.

MZ-based finite memory models relate to adjoint stabilization.

Closure terms resemble upwind flux corrections in linear advection.

Abstract

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.