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||David Shirokoff|
|April 24||Luc Deike, Princeton University||Shahriar Afkhami|
|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: March 28, 2017