Stochastic subgrid scale modeling and structure-preserving flux limiting for hyperbolic systems

The objective of this project is the derivation of physics-compatible stochastic parametrizations for coarse-grained simulations of flow processes whose microscopic behavior is governed by hyperbolic systems of partial differential equations. Using the framework of variational multiscale (VMS) methods, a fine-grid finite element discretization will be averaged to extract stochastic subgrid scale models for the compressible Euler equations and for the shallow water equations. The proposed approach to subgrid upscaling resembles Reynolds-averaged Navier--Stokes (RANS) turbulence modeling. The filtered equations for "slow" coarse-scale components (averages) contain nonlinear terms that depend on "fast" fine-scale components (fluctuations). Replacing these terms with stochastic processes, one obtains a closed-form reduced system of evolution equations for the averages. There is no guarantee that such multiscale approximations are entropy stable and invariant domain preserving. In this project, physical admissibility of simulation results will be ensured using algebraic flux correction (AFC) tools and, in particular, novel convex limiting techniques developed by the principal investigator and his collaborators for standard finite element discretizations of hyperbolic problems. Monolithic AFC approaches make it possible to enforce not only discrete maximum principles but also sufficient conditions of entropy stability by switching to a structure-preserving low-order scheme in critical regions. The limited fluxes of a stochastic VMS method should be entropy dissipative and satisfy all relevant constraints. Existing relationships to entropy/eddy viscosity models will be investigated theoretically and numerically.

Keywords: hyperbolic conservation laws; stochastic homogenization; finite element schemes; variational multiscale methods; algebraic flux correction; entropy stability