The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non- homogeneous fluid and ocean flows. The Dynamically Orthogonal (DO) field equations provide an efficient time-dependent adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier-Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the orthonormal modes that define a basis for the evolving DO subspace, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with Total Variation Diminishing schemes for the advection terms. Other results specific to the DO equations include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size in- stead of quadratic; (ii) symmetric Total Variation Diminishing-based advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify the correctness of our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.

Prof. Pierre Lermusiaux is an invited international external adviser to the European project Stochastic Assimilation for the Next Generation Ocean Model Applications (SANGOMA), which will be providing new developments in data assimilation for future operational forecasting and monitoring systems. He is giving a talk on Bayesian Data Assimilation and Learning of Stochastic Dynamical Models: State, Parameters, also Model Structures at their first annual meeting.

Abstract: In this talk, we present error analysis and numerical results showing optimal convergence and superconvergence properties of the hybridizable discontinuous Galerkin (HDG) methods. First, we study the convection-diffusion equations with variable-degree approximations on nonconforming meshes. Our results hold for any (bounded) irregularity index of the nonconformity of the mesh, and can be extended to hypercubes. Second, we design and analyze the first HDG methods for stationary, third-order linear equations in one-space dimension. 13 methods are analyzed in a unified setting. They all provide superconvergent approximations to the exact solution u and its two derivatives. Numerical results validate our theoretical findings.

Biography: Yanlai Chen is an Assistant Professor of Mathematics at UMass Dartmouth. Before joining UMassD, he was a postdoctoral research associate at Brown University. He received his Ph.D. in Mathematics and M.S. in computer science from University of Minnesota in 2007, and B.S. degree in Mathematics from University of Science and Technology of China (USTC) in 2002. Dr. Chen’s research area is numerical analysis and scientific computing, in particular discontinuous Galerkin finite element methods, hybridizable discontinuous Galerkin methods, reduced basis method and reduced basis element method. His research is currently supported by NSF.