2017/12/27 by Eric Parish, Karthik Duraisamy, Parish, Eric J. +1 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1712.09669
openalex publication_date 2017/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe a paradigm for multiscale modeling that combines the Mori-Zwanzig\n(MZ) formalism of Statistical Mechanics with the Variational Multiscale (VMS)\nmethod. The MZ-VMS approach leverages both VMS scale-separation projectors as\nwell as phase-space projectors to provide a systematic modeling approach that\nis applicable to non-linear partial differential equations. Spectral as well as\ncontinuous and discontinuous finite element methods are considered. The\nframework leads to a formally closed equation in which the effect of the\nunresolved scales on the resolved scales is non-local in time and appears as a\nconvolution or memory integral. The resulting non-Markovian system is used as a\nstarting point for model development. We discover that unresolved scales lead\nto memory effects that are driven by an orthogonal projection of the\ncoarse-scale residual and inter-element jumps. It is further shown that an\nMZ-based finite memory model is a variant of the well-known\nadjoint-stabilization method. For hyperbolic equations, this stabilization is\nshown to have the form of an artificial viscosity term. We further establish\nconnections between the memory kernel and approximate Riemann solvers. It is\ndemonstrated that, in the case of one-dimensional linear advection, the\nassumption of a finite memory and a linear quadrature leads to a closure term\nthat is formally equivalent to an upwind flux correction.\n