SCIENTIFIC PROGRAMS AND ACTIVITIES

Stewart Library

Chris Budd (University of Bath, UK) - 2:00 p.m.
Mesh generation using optimal transport

When numerically solving a PDE in three dimensions , it is often necessary to generate a mesh on which to discretize the solution. Often this can be expensive to do. However, by using ideas from optimal transport it is possible both to construct a mesh quickly and cheaply, and also to prove that it has the necessary regularity properties to allow an accurate approximation of the solution of the PDE. In this talk I will describe these methods, prove results about their regularity and then apply them to some problems in meteorology.

Friday May 8, 2015

Stewart Library

Roderick S. C. Wong (City University of Hong Kong) - 11:00 a.m.
Asymptotics and Orthogonal Polynomials

In this talk, we review some of the methods that are now available in asymptotics, and show how they can be applied to classical and non-classical orthogonal polynomials. These include methods in asymptotic evaluation of integrals and asymptotic theory for ordinary differential equations. Also included are two newer methods, namely, the Riemann-Hilbert method and asymptotics for linear recurrences.

Tuesday May 5, 2015

Stewart Library

Jerome Neufeld (University of Cambridge) - 2:00 p.m.
Dynamics of fluid-driven fracturing

Fluid-driven delamination or fracturing occurs in a host of materials with applications to the damage of biological tissues, the deformation of engineered materials and the fracture and deformation of the Earth’s crust. In direct analogy with the physical and mathematical complexities faced at the contact line of a spreading capillary drop, we show that bending and in-plane tension within the sheet play particularly crucial roles near the delaminating tip of an elastic blister. Matching a quasi-static interior blister to the dynamics as the peeling or pulling tip thereby determining the evolution of a fluid blisters beneath thin elastic sheets.
The macroscopic manifestation of the contact line are highlighted by a new experimental system, elastic magnetic sheets, in which the magnetic attraction which provides a repeatable laboratory analogue in which to study fluid-driven fracturing. This experimental system exhibits transitions from static to dynamically-driven fluid fracturing in both the bending and tensional regimes, and therefore provides a new laboratory analogue in which to repeatably study the dynamics of fluid driven fracturing of elastica.

Lister, J.R., Peng, G.G., Neufeld, J.A. (2013). Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett., 111, 1–5.

 

Wednesday April 8, 2015

Stewart Library

Hiro Oh (University of Edinburgh) - 10:30-11:30 am
Invariant Gibbs measures for Hamiltonian PDEs

In this talk, I will talk about different aspects of invariant Gibbs measures for Hamiltonian PDEs. We first go over the construction of invariant Gibbs measures for Hamiltonian PDEs on the circle due to Bourgain '94. Then, we move onto the real line case. If time permits, I will mention the higher dimensional situation (at least in the periodic setting.)

Oana Pocovnicu (Princeton University) - 12:00-1:00 pm
A modulated two-soliton with transient turbulent regime for a focusing cubic nonlinear half-wave equation on the real line

In this talk we discuss work in progress regarding a nonlocal focusing cubic half-wave equation on the real line. Evolution problems with nonlocal dispersion naturally arise in physical settings which include models for weak turbulence, continuum limits of lattice systems, and gravitational collapse. The goal of the present work is to construct an asymptotic global-in-time modulated two-soliton solution of small mass, which exhibits the following two regimes: (i) a turbulent regime characterized by an explicit growth of high Sobolev norms on a finite time interval, followed by (ii) a stabilized regime in which the high Sobolev norms remain stationary large forever in time. This talk is based on joint work with P. Gerard (Orsay, France), E. Lenzmann (Basel, Switzerland), and P. Raphael (Nice, France).

February 24, 2015
2:00 pm

Stewart Library

Peter Perry (University of Kentucky, Lexington)
The Davey-Stewartson Equation: A Case Study of the Completely Integrable Method in Two Space Dimensions

The Davey-Stewartson II equation is a completely integrable, nonlinear dispersive equation in two space and one time dimensions which describes the amplitude of weakly nonlinear waves in shallow water. We'll discuss recent work on both the defocussing and focussing case which uses the completely integrable method to obtain global well-posedness, and large-time asymptotics (defocussing case) and spectral instability of soliton solutions (focussing case). This work builds on the pioneering work of Ablowitz-Fokas, Beals-Coifman, and others. The technical core of this work is a careful study of a $\overline{\partial}$-problem where the space-time parameters enter through an oscillatory phase.

Yaushu Wong (University of Alberta)
This is a joint work with Kun Wang & Jian Deng
Pollution - free Difference Schemes for Helmholtz Equation in Polar and Spherical Coordinates

The Helmholtz equation arises in many problems related to wave propagations, such as acoustic, electromagnetic wave scattering and models in geophysical applications. Developing efficient and highly accurate numerical schemes to solve the Helmholtz equation at large wave numbers is a very challenging scientific problem and it has attracted a great deal of attention for a long time. The foremost difficulty in solving the Helmholtz equation is to eliminate or minimize the pollution effect which could lead to a serious problem as the wave number increases. Let k, h, and n denote the wave number, the grid size and the order of a finite difference or finite element approximations, we could show that the relative error is bounded by where or 1 for a finite difference or finite element method. Recently, new finite difference schemes are developed for one-dimensional Helmholtz equation with constant wave numbers, and it is shown that error estimate is bounded by and the convergence is independent of the wave number k even when kh>1. In this talk, we extend the idea on constructing the pollution -free difference schemes to multi-dimensional Helmholtz equation in the polar and spherical coordinates. The superior performances of the new schemes are validated by comparing the numerical solutions with those obtained by the standard finite difference and the fourth-order compact schemes.

Two mathematical models will be presented for the lithiation/delithiation of a single nanoparticle in a Lithium-ion battery electrode. The first is a deterministic ODE model that describes quasi-equilibrium and out-of-equilibrium lithiation/delithiation under voltage control. The dynamics of a single particle can be reduced to rapid switching between an ‘empty’ state and a ‘full’ state. Using asymptotic analysis, the critical voltage at which the switch occurs is identified under both static and dynamic conditions.

The second model is based on a probabilistic description of the discrete number of Lithium ions in a single particle. Starting from the chemical master equations, a discrete-to-continuum model is derived for the probability distribution during dynamic charging/discharging. Distinct asymptotic regimes are identified in which either discreteness and thermal noise are important or the dynamics is well captured by the continuous deterministic model.