Date: Friday, January 11, 2019
Location: 1084 East Hall (3:00 PM to 4:00 PM)
Title: Coarsegrained modeling of multiscale PDEs using the MoriZwanzig formalism
Abstract: This talk will address the issue of closure in reduced order models (ROMs) and large eddy simulations (LES), leveraging ideas from nonequilibrium statistical mechanics. The approach is based on the Variational MultiScale method (VMS) and the MoriZwanzig (MZ) formalism, which provides a framework to perform formal scale separation and recast a highdimensional dynamical system into an equivalent, lowerdimensional system. In this reduced system, which is in the form of a generalized Langevin equation (GLE), the effect of the unresolved modes on the resolved modes appears as a convolution integral (which is sometimes referred to as memory). The MZ formalism alone does not lead to a reduction in computational complexity as it requires the solution of the orthogonal dynamics PDE. A model for the memory is constructed by assuming that memory effects have a finite temporal support and by exploiting scale similarity. We discover that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarsescale residual and interelement jumps (in the case of discontinuous finite elements). It is further shown that an MZbased finite memory model is a variant of the wellknown adjointstabilization 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. In the context of ROMs, this model is shown to yield a PetrovGalerkin projection. Several applications in ROMs and LES ranging from simple scalar PDEs to Magnetohydrodynamic turbulence will be presented.
Files:
Speaker: Karthik Duraisamy
Institution: University of Michigan
Event Organizer: Divakar Viswanath divakar@umich.edu
