Seminars are held on Mondays from 2:30 - 3:30PM in Cullimore Hall, Room 611, unless noted otherwise. For questions about the seminar schedule, please contact David Shirokoff.
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|Date||Speaker, Affiliation, and Title||Host|
|March 27||Tiago Salvador, McGill University
Building Accurate Convergent Finite Difference Schemes for Elliptic Partial Differential Equations Abstract
The approximation theory of Barles and Souganidis guarantees the convergence of numerical schemes to the viscosity solution of the underlying partial differential equation (PDE) provided they are monotone, stable and consistent. However, these schemes are in general only 1st order accurate. Recently, Froese and Oberman introduced convergent filtered schemes, which achieve high order where the solutions are sufficiently smooth. In this talk, we will use these tools to build accurate convergent finite difference schemes for elliptic partial differential equations.
We build filtered schemes for Hamilton-Jacobi equations. The schemes are accurate and explicit, and so they are solved using the fast sweeping method. We build a monotone wide stencil finite difference scheme for the 2-Hessian equation. This equation is only elliptic on a restricted set of functions, which makes the discretization challenging. Accurate, but not provably convergent schemes are also considered. Finally, if time allows, we discuss numerical methods for the nonlinear partial differential equation that governs the motion of level sets by affine curvature: we show that standard finite difference schemes are nonlinearly unstable and build a convergent monotone finite difference scheme.
|April 3||Ricardo Barros, Loughborough University
Large Amplitude Internal Waves in Three-Layer Flows
Large amplitude internal waves in a three-layer flow confined between two rigid walls will be examined in this talk. The mathematical model under consideration arises as a particular case of the multi-layer model derived without imposing any smallness assumption on the wave amplitudes and is well-suited to describe internal waves within a strongly nonlinear regime. Solitary-wave solutions will be investigated and some of their properties will be unveiled by carrying out a detailed critical point analysis of the underlying dynamical system. We will also address the role played by criticality on the polarity of interfacial waves and highlight some shortcomings of the Boussinesq approximation.
|April 10||Luiz Faria, MIT
Towards an Asymptotic Theory of Weak Heat Release Combustion
In this talk we present some recent progress in the development of an asymptotic theory of weak heat release combustion encompassing both ames and detonations. In particular, we review and extend the multi-dimensional weakly nonlinear theory of detonations presented in  in order to account for transport effects (i.e. viscosity, heat diffusion, and species diffusion). Employing an extension of the method of matched asymptotic expansion , we derive a system of PDEs whose solutions represent a uniformly valid approximation to the reactive Navier-Stokes equations in the limit considered. By computing the one-dimensional traveling waves, we show that both subsonic and supersonic solutions exist in the simplified model. Finally, we comment on an important drawback of the theory, which automatically excludes very slow waves (Mach << 1), and suggest possible ways to circumvent such limitations.
 L. Faria, A. Kasimov, R. Rosales. Theory of weakly nonlinear self-sustained detonations. Journal of Fluid Mechanics, 2015.
 L. Faria, R. Rosales. Equation level matching: and extension of the method of matched asymptotic expansions for problems of wave propagation. ArXiv, 1701.05882, 2017.
|April 17||Ian Griffiths, University of Oxford
Wrinkles, ripples and deformations arise in a range of situations in everyday life where they are unwanted. In this talk we derive mathematical models that explain the appearance and evolution of wrinkles in glass-sheet manufacture and elastic membranes, both of which are used in electronic devices and smartphones. We show how asymptotic analysis techniques can assist us in smoothing out these ripples. We conclude with an application in which smoothness is the most important feature, to discuss how mathematics can help us with the creation of the perfect fruit smoothie.
|April 24||Luc Deike, Princeton University
Air Entrainment and Bubble Statistics Under Breaking Waves"
Complex turbulent mixture of fluids are encountered in various environmental situations: air water fluxes in the ocean, lakes or in coastal areas, spray dynamics in the atmosphere or water droplets in clouds. They are also prevalent in industrial contexts, such as atomization in chemical reactors, or oil and gas transportation challenges. However a fundamental understanding of the general multi-scale properties of such multiphase turbulent flows is still lacking.
Breaking waves at the water surface is a striking example of turbulent mixing across a fluid interface. The impact of the jet generates turbulence, entrains air into the water and ejects droplets into the air. In this talk, I will present laboratory experiments and novel direct numerical simulations of breaking waves that bring new insights into the associated two-phase turbulent flow. I specifically address the dissipation of energy and the entrainment of air bubbles for a single breaking wave. I will then discuss the up-scaling to the ocean and implications for air-sea exchanges of gases and marine aerosols, key to the climate system.
|May 1||Michael Mueller, Princeton University
Physics-Based Approaches to Model Form Uncertainty Quantification for Large Eddy Simulation of Turbulent Combustion
All models have errors that result in prediction uncertainty. The prevailing approach in model form uncertainty quantification is to calibrate a model “mismatch” or model “inadequacy” term against data, which is then used to determine the prediction uncertainty for a quantity of interest. This data-based approach requires data that may not be available or may not be available over all conditions and fails to take advantage of the model’s underlying physics. In this seminar, an alternative approach to model form uncertainty quantification will be presented that is inherently physics-based. The objective is to translate model assumptions, which ultimately result in model errors, into mathematical statements of uncertainty. In some situations, implicit model assumptions are required due to an ignorance of the underlying physics. To estimate the error in a model, an equally plausible peer model is proposed, with the difference between the two models serving as an estimate for the model error. In other situations, explicit model assumptions are required for the sake of computational expediency. In these cases, a set of hierarchical models can be constructed with nested assumptions. Physical principles in a higher-fidelity model in the hierarchy can be directly used to estimate the error in a lower-fidelity model in the hierarchy.
These two approaches will be leveraged to quantify/estimate the uncertainties arising from different component models in Large Eddy Simulation (LES) of turbulent combustion. The notion of peer models is applied to modeling of unresolved turbulent mixing, and the notion of hierarchical models is applied to the modeling of unresolved combustion processes.
Updated: April 20, 2017