The Department of Applied Mathematics is pleased to host this series of colloquium lectures, funded in part by a generous gift from the Boeing Company. This series will bring to campus prominent applied mathematicians from around the world.
The talks should be of general interest to researchers and students in the mathematical sciences and related fields. All are welcome to attend.
There will be four talks in this series each quarter, on Thursday afternoons at 4:00pm in Gug 220. Each talk will be followed by a reception. Below is a list of the Boeing seminars in the autumn, winter, and spring quarters of the 2011-2012 academic year.
Current Boeing Seminars for 2011-2012
Winter Quarter 2012
January 5, 2012
Marsha Berger, Courant Institute of Mathematical Sciences
Cell-cut methods for flows in complicated geometry
The Cartesian grid embedded boundary approach has attracted much interest in the last decade due to the ease of grid generation for complicated geometries. This approach uses rectangular Cartesian meshes over most of the domain, with irregular, or cut cells at the where the mesh intersects a solid body. It is in routine use in design projects at NASA Ames to automate the solution of steady inviscid compressible flow. Extending the method however to compute time-dependent flow or viscous flow is much more complicated. In this talk we first briefly describe our approach to embedded boundary computations, and illustrate what distinguishes it from level set or other immersed boundary approaches. We present examples showing the current state for computing steady invsicid flow at NASA Ames. The second half of the talk will concentrate on the algorithmic issues that arise when trying to extend the method to compute time-dependent and viscous flows. We present our preliminary work in two space dimensions illustrating a simpli- fied but second-order accurate approach to solving the so-called small-cell problem faced by explicit difference schemes. We also show our recent work in computing high Reynolds number flow without using the anisotropic re- finement which is possible with a body-fitted grid.
January 19, 2012
Walter Strauss, Brown University
Steady Rotational Water Waves
Precise study of water waves began with the derivation of the basic mathematical equations of fluids by the great Euler in 1752. In the two and a half centuries since then, the theory of fluids has played a central role in the development of mathematics. Water waves are fluids with a free surface. I will discuss periodic waves that travel at a constant speed. Using local and global bifurcation theory, we now know how to prove that there exist very many such waves. They may have either small or large amplitudes. I will outline the existence proof, joint with Adrian Constantin, and then exhibit some computations, joint with Joy Ko, of the waves using numerical continuation. The computations illustrate certain relationships between the amplitude, energy and mass flux of the waves. If the vorticity is sufficiently large, the first stagnation point of the wave occurs either at the crest, on the bed directly below the crest, or in the interior of the fluid. The vorticity also affects the pressure beneath the fluid and the behavior of the fluid particles. This work is a perfect example of the synergy between theory and computation.
January 26, 2012
Gigliola Staffilani, Massachusetts Institute of Technology
Dispersive equations and their role beyond PDE
Arguably the star in the family of dispersive equations is the Schrodinger equation. Among many mathematicians and physicists it is regarded as fundamental, in particular to understand complex phenomena in quantum mechanics. But not many people may know that this equation, when defined in a periodic setting for example, has a very reach and more abstract structure that touches several fields of mathematics, among which analytic number theory, symplectic geometry, dynamical systems and probability. In this talk I will illustrate in the simplest possible way how all these different aspects of a unique equation have a life of their own while interacting with each other to assemble a beautiful and subtle picture.
Spring Quarter 2012
Past Boeing Seminars
Autumn Quarter 2011
October 6, 2011:
James Murray, Princeton University/University of Washington
Vignettes from a mathematician’s odyssey in biology: from pilot ejection injuries to the benefits of cannibalism
Abstract: I shall describe a series of problems for which the models are both practical and the mathematics simple. Pilot ejection can cause extreme stresses on the spine which frequently result in major injuries. A basic model of the physical process shows how such injuries could occur. Many diseases require a careful assessment of medication level: liver disease is a prime example. I shall describe the medical dilemma and show how the model provided a scientific method for determining the appropriate medication level. It is now used in England. Breathalysers are notoriously inaccurate. I shall describe two models which show that it is not possible to design a better breathalyzer. Prostate specific antigen (PSA) blood tests are considered indicators of possible prostate cancer but are notoriously misunderstood and misinterpreted. Our model explains a known anomaly and highlights inadequacies in current government guidelines. Cannibalism is widespread in nature and generally of benefit to the species. I shall describe a model for salamander cannibalism and then describe how important cannibalism has been for humans. Finally I shall show how certain spatial patterns in development are influenced by geometry and scale and, time permitting, how the woods on Bainbridge Island can exhibit jungle like aspects.
October 27, 2011:
Eliot Fried, University of Washington
Some features and challenges of the Navier-Stokes-alpha-beta equation
Abstract: The Navier-Stokes-alpha-beta equations regularize the Navier-Stokes equations by the addition of dispersive and dissipative terms. The dispersive term is proportional to the divergence of the corotational rate of the symmetric part of the velocity gradient. The dissipative term is proportional to the bi-Laplacian of the velocity. The coefficients of these terms involve factors alpha and beta, respectively, both having dimensions of length. Calculating the energy spectrum for an assembly of stretched spiral vortices reveals an inertial range where Kolmogorov's -5/3 law holds and shows that choosing beta less than alpha yields a better approximation of the inertial range of the Navier-Stokes equations. Direct numerical simulations of three-dimensional periodic turbulent flow confirm this and also show that vorticity structures behave more realistically when beta is less than alpha. However, the simulations indicate that optimal choices of alpha and beta are resolution dependent. This suggests the possibility of developing multigrid methods that capitalize on resolution dependence by using the Navier-Stokes-alpha-beta equations at coarse grid levels, with different choices of alpha and beta at each level, to accelerate convergence to solutions of the Navier-Stokes equation at the finest grid level. Results obtained from a two-dimensional spectral multigrid algorithm of this type show promise.
November 10, 2011:
Felix Otto, University of Bonn
Pattern formation and Partial Differential Equations
Abstract: In three specific examples, we shall demonstrate how the theory of partial differential equations (PDEs) relates to pattern formation in nature: Spinodal decomposition and the Cahn-Hilliard equation, Rayleigh-B\'enard convection and the Boussinesq approximation, rough crystal growth and the Kuramoto-Sivashinsky equation. These examples from different applications have in common that only a few physical mechanisms, which are modeled by simple-looking evolutionary PDEs, lead to complex patterns. These mechanisms will be explained, numerical simulation shall serve as a visual experiment. Numerical simulations also reveal that generic solutions of these deterministic equations have stationary or self-similar statistics that are independent of the system size and of the details of the initial data. We show how PDE methods, i. e. a priori estimates, can be used to understand some aspects of this universal behavior. In case of the Cahn-Hilliard equation, the method makes use of its gradient flow structure and a property of the energy landscape. In case of the Boussinesq equation, a ``driven gradient flow'', the background field method is used. In case of the Kuramoto-Sivashinsky equation, that mixes conservative and dissipative dynamics, the method relies on a new result on Burgers' equation.