# Applied Mathematics Colloquium - Spring 2017

## Colloquium Schedule

Colloquia are held on Fridays at 11:30 a.m. in Cullimore Lecture Hall II, unless noted otherwise. Refreshments are served at 11:30 am. For questions about the seminar schedule, please contact Yassine Boubendir.

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Date | Speaker, Affiliation, and Title | Host |
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January 20 | Brittany Froese, NJITMeshfree Finite Difference Methods for Fully Nonlinear Elliptic Equations The relatively recent introduction of viscosity solutions and the Barles-Souganidis convergence framework have allowed for considerable progress in the numerical solution of fully nonlinear elliptic equations. Convergent, wide-stencil finite difference methods now exist for a variety of problems. However, these schemes are defined only on uniform Cartesian meshes over a rectangular domain. We describe a framework for constructing convergent meshfree finite difference approximations for a class of nonlinear elliptic operators. These approximations are defined on unstructured point clouds, which allows for computation on non-uniform meshes and complicated geometries. Because the schemes are monotone, they fit within the Barles-Souganidis convergence framework and can serve as a foundation for higher-order filtered methods. We present computational results for several examples including problems posed on random point clouds, computation of convex envelopes, obstacle problems, non-continuous surfaces of prescribed Gaussian curvature, and Monge-Ampere equations arising in optimal transportation. |
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January 27 | Andrew Comech, Texas A&M UniversityStability of Solitary Waves in the Nonlinear Dirac Equation We discuss stability of solitary waves in the nonlinear Schroedinger equation and in nonlinear Dirac equation. We prove that weakly relativistic solitary waves in thenonlinear Dirac equation are spectrally stable in the case of NLS-subcritical nonlinearity: that is, the linearization at a solitary wave has purely imaginary spectrum. |
Yassine Boubendir |

February 3 | Douglas Wright, Drexel UniversityTraveling Waves in Diatomic Fermi-Pasta-Ulam-Tsingou Lattices Consider an infinite chain of masses, each connected to its nearest neighbors by a (nonlinear) spring. This is an FPUT lattice. In the instance where the masses are identical, there is a well-developed theory on the existence, dynamics and stability of solitary waves and the system has come to be one of the paradigmatic examples of a dispersive nonlinear equation. In this talk, I will discuss recent rigorous results of mine (together with T. Faver, A. Hoffman, R. Perline, A. Vainchstein and Y. Starosvetsky) on the existence of traveling waves in the setting where the masses alternate in size. In particular I will address in the limit where the mass ratio tends to zero. The problem is inherently singular and as such the existence theory becomes rather complicated. In particular, we find that the traveling waves are not true solitary waves but rather "nanopterons," which is to say, waves which asymptotic at spatial infinity to very small amplitude periodic waves. Moreover, we can only find solutions when the mass ratio lies in a certain open set. The difficulties in the problem all revolve around understanding Jost solutions of a nonlocal Schrodinger operator in its semi-classical limit. |
Roy Goodman |

February 10 | Mark Hoefer, University of Colorado at BoulderDispersive Hydrodynamics Localized traveling wave solutions termed solitons are one of the most prominent and admired features of nonlinear, dispersive partial differential equations. Recent years have witnessed a rising interest, both mathematically and physically, in another class of solutions: dynamically expanding, oscillatory nonlinear wave structures called dispersive shock waves (DSWs). A DSW can be viewed as the spatially extended counterpart of a soliton and the dissipationless counterpart of a viscous shock wave. Just like their soliton cousin, DSWs have been observed in a wide range of physical wave systems from geophysical fluids and superfluids to intense light propagation. From soliton-DSW hydrodynamic tunneling to non-classical DSWs in non-convex media, this talk will present some of the latest mathematical and experimental results on DSWs. |
Richard Moore |

February 17 | Sean Sun, Johns Hopkins UniversityWater Dynamics in Cell Mechanics Water plays a fundamental role in many aspects of cell dynamics. From fundamental considerations as well as recent microfluidic experiments, it is now clear cell shape changes and cell movement can be driven by water as well as actomyosin dynamics. The water-driven mode of cell migration is mediated by osmotic pumping of ions (osmotic engine model) across the cell membrane, and is especially important in cancer metastasis. It is not clear what determines the relative contributions of actin and water to the observed cell movement. Here, we develop a 2-phase hydrodynamic model of cell migration, incorporating actin flow and water flow, frictional forces from cell adhesions, and movement of the cell membrane. When all of the mechanical forces are considered, we find that the frictional resistance experienced by the migrating cell can determine the relative contribution of actin and water to the observed cell speed. More generally, osmotic regulation is an important aspect of cell mechanical behavior, and determines slow deformation mechanics of tissue cells. We find that there is a feedback control system that couples membrane tension to actomyosin contraction, which explains how cells respond to mechanical forces. |
Yuan-Nan Young |

February 24 | Ruben Rosales, MITThe Correction Function Method [CFM] for Interfaces/Boundaries Embedded in Regular Grids Computations that involve moving interfaces across which the variables (and their gradients) jump are very important in practice. In particular, embedded interface methods, which do not require re-meshing as the interfaces move, are very attractive, but hard to push into high orders. Two well known examples are the Immersed Boundary Method [IBM], and the Immersed Interface Method [IIM]. In this talk I will introduce a close cousin to these methods, using the example of the Poisson equation. The objective is to develop an approach that allows a systematic road to high order approximations. The Poisson equation with jumps in function value and normal derivative across an interface is of central importance in Computational Fluid Dynamics. The basic idea of the CFM is similar to the Ghost Fluid Method [GFM], which relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the right hand side [RHS] of the equations, producing a problem with the same linear system as if there were no jumps, but different RHS. If not, the result is a modified linear system which has the same sparsity pattern as the standard discretization. However, achieving high accuracy is not simple with the "standard" approaches used to compute the GFM correction terms. In the CFM the corrections are computed via a "correction function", defined in a narrow band around the interface via a pde with with appropriate boundary conditions. This pde can, in principle, be solved to any desired order of accuracy. For example, a solution compatible with 4-th order accuracy can be obtained by representing the correction function in terms of Hermite interpolants (bi-cubics in 2-D), followed by an appropriate least square minimization. Higher order accuracy follows, in principle, by using higher order interpolants, with the same approach. In this talk I will first present the CFM method for the simplest Poisson problem where the jumps in the function values and derivatives have the same weights. Then I will describe how to implement the method for the general Poisson problem, where several alternatives are possible. Finally, if time permits, I will show the results of some preliminary work on how to extend the method for other pde problems. |
David Shirokoff |

March 3 | Yuriko Renardy, Virginia TechA Viscoelastic Model for Thixotropic Yield Stress Fluids: Shear, Elongation, and Shear Banding This is an overview of work using a partially extending strand convection model for an entangled microstructure, within a Newtonian solvent (PECN), for homogeneous parallel shear flow, uniaxial and biaxial elongation, filament stretching, and shear banding. Thixotropy refers to an apparent viscosity which depends on the time since the material last flowed. For the PECN model, thixotropy is characterized by a long relaxation time of the microstructure and a non monotone shear stress - shear rate curve for steady solutions. This presentation includes joint works with Kara Maki (RIT), Holly Grant and Michael Renardy (Virginia Tech). |
Shahriar Afkhami |

March 10 | Diane Henderson, Pennsylvania State UniversityA Method to Recover Water-Wave Profiles from Bottom Pressure Measurements A fully nonlinear, closed-form mapping from time series of pressure measurements at an arbitrary depth in the fluid column to time series of surface displacement is derived from the full Stokes boundary value problem for steady water waves. The formula is implicit and nonlocal and must be solved numerically, albeit to machine precision. We further use it to derive several explicit, asymptotic formulae and compare the fully-nonlinear formula, the asymptotic formulae, and the two main formulae presently used by engineers to numerically generated exact solutions of Euler’s equations and to our laboratory experiments on solitons, wave groups, periodic (cnoidal) waves and reflected waves. (Diane Henderson with Bernard Deconinck, Katie Oliveras, Vishal Vasan) |
Wooyoung Choi |

March 24 | Linda Cummings, NJITModeling and Simulation of Thin Film Flows of Nematic Liquid Crystal The Leslie-Ericksen theory for nematic liquid crystals will be used to derive a model for free surface flow of a thin film of NLC. Formal asymptotics based on the film aspect ratio lead to a 4th order nonlinear PDE governing the film height evolution. The model will be studied and simulated in different regimes, where different physical effects can lead to instability. Our large scale simulations for ultra-thin films where van der Waals' forces are important show remarkable qualitative agreement with experimental observations. |
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March 31 | Steven G. Johnson, MITBounds on Light-Matter Interactions It is now widely recognized that, by taking ordinary materials and rearranging them into complex shapes on the same scale as the wavelength of light (or other wave-propagation systems), an amazing variety of new phenomena are possible. This has been exploited to engineer new classes of optical devices, from ultra-efficient solar cells to exotic optical fibers, as well as new phenomena of interest in basic physical research. A key challenge is the enormous number of degrees of freedom available to modern nanofabrication, combined with a lack of closed-form analytical solutions in all but the most trivial geometries. Instead, one form of analytical guidance comes in the form of theoretical bounds on attainable performance, which serve as both constraints and as targets to meet, or even exceed (by circumventing the assumptions underlying the bounds). Many famous such results include the Yablonovitch limit for solar cells, the Manley-Rowe limits to nonlinear frequency conversion, and the Wiener bounds on homogenized material properties. In this talk, I will review these along with some newer bounds derived in our group: scaling limitations on cloaking, upper bounds on scattering/absorption, upper bounds on the local density of states and radiation, and limits on the performance of "exceptional" resonances. Several of the bounds are derived surprisingly simply from fundamental energy considerations: the absorbed power (a quadratic function of the fields) is bounded above by the total absorbed+scattered power (the real part of a linear function of the fields via the optical theorem), and the resulting quadratic |
David Shirokoff |

April 7 | Kresimir Josic, University of Houston |
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April 14 | Bob Pego, Carnegie Mellon University |
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April 21 | Robert Krasny, University of Michigan at Ann Arbor |
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April 28 | He Xiaoming, Missouri University of Science & TechnologyDual-Porosity Stokes Model and Finite Element Method for Coupling Dual-Porosity Flow and Free Flow We propose and numerically solve a new model considering confined flow in dual-porosity media coupled with free flow in embedded macro-fractures and conduits. Such situation arises, for example, for fluid flows in hydraulic fractured tight/shale oil/gas reservoirs. The flow in dual-porosity media, which consists of both matrix and micro-fractures, is described by a dual-porosity model. And the flow in the macro-fractures and conduits is governed by the Stokes equation. Then the two models are coupled through four physically valid interface conditions on the interface between dual-porosity media and macro-fractures/conduits, which play a key role in a physically faithful simulation with high accuracy. All the four interface conditions are constructed based on fundamental properties of the traditional dual-porosity model and the well-known Stokes-Darcy model. The weak formulation is derived for the proposed model and the well-posedness of the model is analyzed. A finite element semi-discretization in space is presented based on the weak formulation and four different schemes are then utilized for the full discretization. The convergence of the full discretization with backward Euler scheme is analyzed. Four numerical experiments are presented to validate the proposed model and demonstrate the features of both the model and numerical method, such as the optimal convergence rate of the numerical solution, the detail flow characteristics around macro-fractures and conduits, and the applicability to the real world problems. |
Yassine Boubendir |

*Updated: March 28, 2017*