All seminars (unless otherwise stated) will take place on **Tuesdays at 3.00pm in ****Room 505 in the Mathematics Department **(25 Gordon Street). See see how to find us for further details. There will be tea afterwards in Mathematics Room 606. If you require any more information on the Applied seminars please contact Prof Slava Kurylev e-mail: y.kurylev AT ucl.ac.uk or tel: 020-7679-7896.

## 29 September 2015 at 3pm

Dr Anatasia Kisil, University of Cambridge

##### Title: Approximate matrix Wiener-Hopf factorisation and applications to problems in acoustics

**Abstract:** This talk will introduce a technique for solving a class of PDEs called the Wiener-Hopf method. It is an elegant method which extends the separation of variables technique used to investigate PDEs. After introducing the topic I will talk about my work on the approximate matrix Wiener-Hopf factorisation and stability analysis. This is then applied to the scattering of a sound wave by an infinite periodic grating composed of rigid plates (joint work with I. D. Abrahams).

## 6 October 2015 at 3pm

Dr Yue-Kin Tsang, University of Exeter

##### Title: Advection-condensation of water vapour in a stochastic model with coherent stirring

**Abstract:** Atmospheric water vapour is an important greenhouse gas and has a strong impact on climate. Observations show that large-scale moisture dynamics can be captured by kinematic models in which water vapour is treated as a passive scalar advected by a prescribed flow and reacts through condensation. Condensation acts as a sink that maintains specific humidity below a prescribed, spatially dependent saturation value. Models of this type have previously been studied using spatially decorrelated Brownian motion as a crude approximation to turbulent motion. In this work we examine the effect of a coherent flow represented by a single vortex which adds a deterministic component to the random motion of fluid parcels. We consider two problems: (i) the drying of a water-vapour anomaly released in the flow at an initial time, and (ii) the steady-state water-vapour distribution achieved in the presence of a moisture source at a boundary. In the strong vortex limit, we solve the governing stochastic differential equations and derive the distribution of specific humidity. Monte Carlo simulations are used to verify these theoretical results and to explore the system beyond the strong vortex limit.

## 13 October 2015, 4pm

Prof Paul Houston, School of Mathematical Sciences, University of Nottingham, UK

##### Title: Adaptive Finite Element Methods for PDEs Posed on Complicated Domains.

**Abstract:** In this talk we consider high-order/hp-version interior penalty discontinuous Galerkin methods for the discretization of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements. By admitting such general meshes, this class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or micro-structures. While standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying (composite) finite element space based on general polytopic meshes is independent of the number of geometric features. Here we consider both the a priori and a posteriori error analysis of this class of methods, as well as their application within Schwarz-type domain decomposition preconditioners.

## 20 October 2015 at 4-5pm

Dr Garth Wells, University of Cambridge

**Location of Seminar: Harry-Massey Lecture Theatre**

##### Title: High-performance differential equation solvers made easy

**Abstract:** There is typically a disconnect between the tools used by domain scientists and research engineers exploring scientific problems through simulation, and the leading edge of high-performance solver technology. Bridging this gap is important to accelerate the exploitation of new computational and mathematical tools for enabling new scientific investigations and new engineering design. In the context of the FEniCS Project (<http://fenicsproject.org>), we have been successful in making the creation of new, performant finite element-based solvers compact and expressive, and accessible and straightforward for domain scientists. A natural step to is to place a wide range of high-performance solvers at the fingertips of researchers, allowing them explore the parameter space of sophisticated solvers for advanced applications with ease, with the objective of finding scalable, parallel solvers for new and changing models. I will present a number of examples of how this can be made easy, from elliptic problems with over 10 billion degrees of freedom, to multi-field equations, to some of the largest practical engineering simulations performed.

## 27 October 2015 at 3pm

Prof Yury Stepanyants, University of Southern Queensland, Australia

##### Title: Analysis of modulational stability of quasi-harmonic wave trains in media with double dispersion (with S P Nikitenkova, N N Singh)

**Abstract:** The problem of modulation stability of quasi-monochromatic long waves propagating in a medium with the double dispersion is revised. The nonlinear Schrödinger equation which describes the evolution of narrow-band wave-trains is derived from the primitive set of shallow-water equations for the rotating fluid in the Boussinesq approximation. The regions of stability and instability in the spectral space are obtained for various signs of dispersion coefficients. Application of results to concrete physical systems (water waves, plasma waves, waves in solids, etc.) is discussed.

## 3 November 2015 - Colloquium Talk

Prof John Willis - DAMTP, University of Cambridge

##### - please see the Departmental Colloquia webpage

## 10 November 2015 - No Seminar (Reading Week)

## 17 November 2015 at 3pm

Dr Rodolphe Sepulchre, University of Cambridge, Department of Engineering

##### Title: Differentially positive system

**Abstract:** Differentially positive systems are systems whose linearization along an arbitrary trajectory is positive. The geometric picture is that a given cone field is infinitesimally contracted along the flow. This property induces a (conal) order that strongly constrains the asymptotic behavior of solutions.

The talk will introduce the concept on simple examples and motivate its relevance for the analysis of nonlinear behaviors in engineering. The results illustrate that behaviors constrained by local order properties extend beyond the well-studied class of linear positive systems and monotone systems, which both require a constant cone field and a linear state space. In particular, differential positivity suggests a novel approach to analyze limit cycles in possibly high-dimensional systems.

Joint work with Fulvio Forni. Preprint on arxiv: http://arxiv.org/abs/1405.6298

## 24 November 2015 at 3pm

Dr H Salman, University of East Anglia

##### Title: Long-range Ordering and Negative Temperature States of Quantized Vortices in a Two-dimensional Superfluid

**Abstract: **We study the relaxation of a 2D superfluid from a non-equilibrium initial state consisting of vortices with positive and negative circulation in experimentally realizable square and rectangular traps. We focus on how like-signed quantized vortices can form clusters and show that such clustering can be understood in terms of negative temperature states of a vortex gas. Using a mean field approximation for the vortex gas, we identify an order parameter that is related to the formation of long-range correlations between the vortices. It turns out that the order parameter corresponds to the stream function of a 2D flow field that is governed by a Boltzmann-Poisson equation. It is, therefore, associated with the emergence of a mean rotational hydrodynamic flow with a nonzero coarse-grained vorticity field. Solutions of the Boltzmann-Poisson equation in a square domain reveal that maximum entropy states of the vortex gas correspond to a large scale monopole flow field. A striking feature of this mean flow, is the spontaneous acquisition of angular momentum by a superfluid flow with a neutral vortex charge. These mean-field predictions are verified through direct simulations of a point vortex gas and 2D simulations of the Gross-Pitaevskii equation. Due to the long-range nature of the Coulomb-like interactions in point vortex flows, the negative temperature states strongly depend on the shape of the geometry. By modifying the domain to a rectangular region, we identify a geometry induced phase transition of the most probable mean flow field which our numerical simulations reproduce

## 1 December 2015 at 3-4pm

Dr Peter Stewart, School of Mathematics and Statistics, University of Glasgow

##### Title: Microstructural effects in aqueous foam fracture

**Abstract:** We examine the fracture of a quasi two-dimensional aqueous foam under an applied driving pressure, using a network modelling approach developed for metallic foams . In agreement with experiments, we observe two distinct mechanisms of failure analogous to those observed in a crystalline solid: a slow ductile mode when the driving pressure is applied slowly, where the void propagates as bubbles interchange neighbours through the T1 process, and a rapid brittle mode for faster application of pressures, where the void advances by successive rupture of liquid films driven by Rayleigh--Taylor instability.

This is joint work with Prof S Hilgenfeldt, Mechanical Sciences and Engineering, University of Illinois, Urbana-Champaign, IL

## 8 December 2015 at 3pm

Dr Ory Schnitzer, Imperial College London

##### Title: Singular asymptotics of surface-plasmon resonance

**Abstract:** Surface plasmons are collective electron-density oscillations at a metal-dielectric interface. In particular, surface-plasmon modes of nano-metallic structures with narrow gaps, which enable a tuneable resonance frequency and a giant near-field enhancement, are at the heart of numerous nanophotonics applications. In this work, we elucidate the singular near-contact asymptotics of such structures. In the classical regime, valid for gap widths > 1nm, we find a generic scaling describing the drastic redshift of the resonance frequency as the gap width is reduced, and in several prototypical dimer configurations derive explicit expressions for the plasmonic eigenvalues and eigenmodes using matched asymptotics; we also derive expressions describing the resonant excitation of such modes by light based on a weak-dissipation limit. In the sub-nanometric ``nonlocal'' regime, we show intuitively and by systematic analysis of the hydrodynamic Drude model that nonlocality manifests itself as a potential discontinuity, and in the near-contact limit equivalently as a widening of the gap. We thereby find the near-contact asymptotics as a renormalisation of the local asymptotics, and in particular a lower bound on plasmon frequency, scaling with the 1/4 power of the Fermi wavelength. Joint work with Vincenzo Giannini, Richard V. Craster and Stefan A. Maier

## 15 December 2015 at 3pm

Dr Danila Prikazchikov, Keele University

##### Title: Low-frequency vibrations of strongly inhomogeneous composite elastic structures

**Abstract: **Low-frequency analysis of strongly inhomogeneous elastic structures is performed. The 1D toy problem for a 3-component elastic rod reveals the effect of high contrast on natural vibrations in the low-frequency region. In particular, the conditions on material and geometric parameters, leading to first eigenfrequency tending to zero, are discussed.

The analysis is then extended to a 3-layered strongly inhomogeneous elastic plate. We concentrate on the practically motivated setups in which the lowest non-zero thickness shear resonance is close to zero. In this case the associated dispersion relation exhibits two low-frequency vibration modes instead of a single fundamental low-frequency mode typical for a homogeneous plate. A possibility of two-mode low-frequency asymptotic models for three-layered plate bending is discussed. The cases, in which a uniformly valid two-mode theory cannot be constructed, are addressed. Comparisons with engineering Timoshenko-type theories are presented.