Computational complex analysis for free surface flows and other applications
Monday 20 April–Wednesday 22 April, 2015
Room 505, Department of Mathematics, University College London
Organizers: Peter Clarkson (Kent), Darren Crowdy (Imperial), Thanasis
Fokas (Cambridge), Robb McDonald (UCL), Beatrice Pelloni (Reading)
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Monday
13:30–13:40
13:40–14:40
14:40–15:10
15:10–15:40
15:40–16.10
16.10–17:10
17:10–18:30
Robb McDonald: Welcome and introduction
Jean-Marc Vanden-Broeck: Numerical studies of new types of
water waves
Michael Dallaston: Numerically determined extinction behaviour
of a bubble in a Hele-Shaw cell
Coffee break
Peter Buchak: Drawing of microstructured optical fibres with
elliptical channels
Michael Siegel: Conformal mapping techniques applied to interfacial flow computations
Wine and cheese reception, room 502
Tuesday
9:30–10:30
10:30–11:00
11:00–11:30
11:30–12.00
12.00–12:30
12:30–14:00
14:00–15:00
15:00–15:30
15.30–16:00
16:00–16:30
16:30–17:00
18:00–
Bengt Fornberg: Numerical Computations on the Painlevé equations
Ana Loureiro: Unique positive solution for the alternative discrete Painleve I equation
Coffee break
Rod Halburd: Singularity and Integrability: Beyond the
Painlevé Property
Alexander Minakov: Riemann-Hilbert problem to the CamassaHolm equation: long-time dynamics of a step-like initial data
Lunch, room 502
Bernard Deconinck: Computing with Riemann surfaces and algebraic curves: the Riemann constant vector
Olga Trichtchenko: Stability of capillary-gravity and flexuralgravity water waves
Coffee break
Dave Hewett: Mathematics of the Faraday cage
Ted Johnson:
Dinner
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Wednesday
9:30–10:30
10:30–11:00
11:00–11.30
11.30–12:30
12:30–14:00
Beatrice Pelloni: A plethora of applications of the Unified Transform
Timo Betcke: Fast and exponentially accurate computation of
scattering from polygons in two dimensions
Coffee break
Tamara Grava: Universality behaviour in Hamiltonian PDES
Lunch (room 502) and close
The Department of Mathematics is located on floors 5-8 of 25 Gordon Street
(above UCL Students Union), at the corner of Gower Place and Gordon
Street. See
http://www.ucl.ac.uk/maths/find-us
http://www.ucl.ac.uk/maps/downloads/ucl-bloomsbury-campus-map-2014
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Abstracts
Timo Betcke (UCL): Fast and exponentially accurate computation of scattering from polygons in two dimensions
In this talk we demonstrate and analyse a method to solve exterior scattering problems from polygonal domains in two dimensions with exponential
accuracy. The method is based on a careful choice of singular basis functions to represent the total field in the presence of corner singularities, and a
representation of the far-field conditions using a first-kind integral equation
approach. The analysis uses Vekua theory to transform the problem into a
standard polynomial approximation problem in the complex plane and allows
the direct computation of the overall exponential convergence rate. Several
numerical examples will be presented including a cavity problem and multiple
scattering configurations.
Peter Buchak (Imperial): Drawing of microstructured optical fibres with
elliptical channels
The advent of microstructured optical fibres (MOFs) has opened up possibilities for controlling light not available with conventional fibre. A MOF,
which differs from a conventional fibre by having an array of channels running along its length, is fabricated by drawing a molten glass preform at low
Reynolds number. However, because surface tension causes the cross section
to deform, the configuration of the channels in the MOF may differ from
the preform. This unintended deformation is inadequately understood and
is difficult to investigate experimentally. As a result, fabricating a desired
configuration may necessitate extensive trial and error.
In this talk, we present a model for the deformation of MOFs with elliptical channels. Our model circumvents the need for expensive computational
methods. More importantly, it can be used to determine the preform configuration required to produce a fibre with a desired arrangement of channels.
We describe comparisons with numerics and experiment and show software
tools that can be used by fabricators to design preforms.
Michael Dallaston (Imperial): Numerically determined extinction behaviour
of a bubble in a Hele-Shaw cell
In this talk I describe a numerical scheme for the evolution of a contracting bubble in a Hele-Shaw cell, including both surface tension (curvaturedependent) and kinetic undercooling (velocity-dependent) terms in the free
boundary condition. The numerical scheme is a spectral collocation method
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applied to the Polubarinova-Galin (PG) equation for Hele-Shaw flow. By
rescaling the scheme in both space and time, we can ascertain the limiting
extinction shape that a bubble approaches as it contracts to a point. In
this small-bubble limit, the Hele-Shaw flow problem reduces to a generalisation of the curve-shortening flow rule. Numerical solutions show that the
generalised flow rule exhibits a far richer array of possible extinction shapes
(circles, lines, and nontrivial n-fold symmetric shapes) than simple curveshortening flow, in which a curve always becomes circular in the limit that
it contracts to a point.
Bernard Deconinck (Washington): Computing with Riemann surfaces and
algebraic curves: the Riemann constant vector
I will present the progress of an ongoing research program to allow for
the efficient computation on Riemann surfaces represented by plane algebraic
curves of arbitrary degree and with arbitrary singularities. Such representations arise naturally when working with solutions of integrable equations
that are given in terms of Abelian functions (generalizations of elliptic functions to more than one variable). Most recently, we have devised algorithms
for the computation of the so-called Riemann constant vector, an object of
great intrinsic importance to a Riemann surface. I will demonstrate the most
updated version of our python program suite abelfunctions, which surpasses
the capabilities of the older maple package algcurves. This is joint work with
Chris Swierczewski.
Bengt Fornberg (Colorado): Numerical Computations on the Painlevé
equations
The six Painlevé transcendents PI to PV I have both applications and
analytic properties that make them stand out from most other classes of
special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation of being
numerically challenging, in particular due to their extensive pole fields in
the complex plane. We note in this present work that, on the contrary, the
Painlevé property (together with the pole fields) in fact provide excellent
opportunities for very fast and accurate numerical solutions across the full
complex plane. In the standard case with solutions real-valued along the real
axis, the complete solution spaces for the PI , PII and PIV equations (featuring 2, 3, and 4 free parameters, respectively) have now been ’surveyed’ in
terms of their pole field dynamics and some other notable solution features.
The present work was carried out in collaborations with André Weideman
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(University of Stellenbosch) and Jonah Reeger (US Air Force Institute of
Technology).
Tamara Grava (SISSA, Trieste): Universality behaviour in Hamiltonian
PDES
We study the critical behaviour of several classes of Hamiltonian PDE
showing that near critical points the behaviour of solutions is described in a
universal way by particular solutions of ODE of the Painlevé family.
Dave Hewett (Oxford): Mathematics of the Faraday cage
Everybody has heard of the Faraday cage effect, whereby a wire mesh
serves to block electric fields and electromagnetic waves. Remarkably, despite 180 years having passed since Faraday originally reported his experimental results, there does not seem to exist in the scientific literature any
satisfactory mathematical analysis of how the strength of the shielding effect
depends on the basic properties of the cage, namely the mesh spacing and
wire thickness. In this talk I will describe some of our recent investigations
into this beautiful problem, using a range of different mathematical tools
including multipole expansions, conformal mappings, integral equations and
numerical quadrature, energy functional minimization, and continuum approximation by the method of multiple scales. This is joint work with Jon
Chapman and Nick Trefethen.
Rod Halburd (UCL): Singularity and Integrability: Beyond the Painlevé
Property
It has long been known that the singularity structure of solutions of an
differential equation in the complex domain gives a strong indicator of its
integrability. Equations with the Painlevé property, that all solutions are
single-valued about all movable singularities, are particularly important in
this respect. I will discuss a number of generalisations of this property.
Methods for finding all solutions of ODEs such that all movable singularities are poles will be discussed, even when the equation itself is not integrable.
This involves a more delicate global analysis of particular solutions than is
the case in Painlevé analysis. In standard Painlevé analysis it is enough to
find movable branching in any solution in order to discard an equation.
If the fixed singularities are also no worse than poles then Nevanlinna
theory provides the necessary global tool. In this talk, new methods will be
applied so as to allow for branching at fixed singularities. Methods for classifying solutions with movable branch points such that globally the number of
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sheets is bounded will be described. The singularity structure of equations
of hydrodynamic type will also be discussed.
Ted Johnson (UCL):
TBA
Ana Loureiro (Kent):Unique positive solution for the alternative discrete
Painleve I equation
The discussion will be about particular solutions for the alternative discrete Painleve I (dP-I) equation and the link to the recurrence coefficients of
certain semiclassical orthogonal polynomials. In fact, it will be shown that
the dP-I equation has a unique solution which remains positive. Furthermore,
such positive solution will be identified as a special solution of Painleve II
involving Airy functions.
Alexander Minakov (Czech Technical University in Prague): RiemannHilbert problem to the Camassa-Holm equation: long-time dynamics of a
step-like initial data
We consider the Camassa-Holm equation ut −utxx +2ux +3uux = 2ux uxx +
uuxxx on the line in the class of functions with strictly positive momentum
variable u−uxx +1 > 0. The initial data is a step-like function, i.e. u(x, 0) →
0 as x → +∞ and u(x, 0) → c as x → −∞, where c ∈ (0, 1) is a positive real
number.
The goal is to reformulate the Cauchy problem as a vector RiemannHilbert problem in view of its further application to study the asymptotic
behavior of the solution of the initial-value problem as t → ∞.
Using the steepest descent method and the so-called g-function mechanism, we deform the originally oscillatory vector Riemann-Hilbert problem to
explicitly solvable model forms and show that the xt− half-plane is divided
into 5 sectors with qualitatively different asymptotic behavior of the solution
of the initial-value problem: a soliton region, a region of a modulated elliptic
wave, two regions of slowly decaying to the constant c self-similar waves, and
a region of fast convergent to the constant c wave.
Beatrice Pelloni (Reading): A plethora of applications of the Unified Transform
Over the past 15 fifteen years, after Fokas introduced it in the late 90s, the
Unified Transform has been shown to be a very flexible and powerful method
for studying a variety of linear and nonlinear problems. The approach was
originally introduced to understand and solve boundary value problem for
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several integrable models in fluid dynamics, but it led to new ideas and a
methodology, based on exploiting analyticity, for studying a much wider pool
of questions. I will discuss the general ideas of this approach, and give a few
examples of specific applications in the study of boundary value problems.
Michael Siegel (NJIT): Conformal mapping techniques applied to interfacial flow computations
The application of conformal mapping techniques to two types of interfacial flow computations will be described. In the first, I will explain how
conformal mapping has been particularly useful in developing and validating
a hybrid asymptotic-numerical method for two-phase flow with soluble surfactant in the practically important limit of large bulk Peclet number. The
method is designed to resolve a narrow transition layer adjacent to the interface across which the surfactant concentration varies rapidly, which presents
severe difficulties for traditional numerical methods. Several challenging examples are presented. In the second part, I will describe results for the flow
induced deformation of a 2D elastic membrane. Here, a nonlinear straining flow can cause the development of finite-time cusp singularities that are
shown to be asymptotically self similar just before cusp formation. The
numerical method tracks the similarity scaling over a 13 decade growth in
the curvature, making this a compelling example of finite-time singularity
formation for interfacial flow.
Olga Trichtchenko (UCL): Stability of capillary-gravity and flexural-gravity
water waves
In this talk, we will show how the equations for periodic, travelling
capillary-gravity as well as flexural-gravity water waves can be reformulated
in terms of a local and a non-local equation. Then, we will examine the stability of the steady-state solutions of these equations by using Hill’s method.
We will finish by comparing and contrasting the two systems and relating to
previously known results.
Jean-Marc Vanden-Broeck (UCL): Numerical studies of new types of water waves
Water waves have been studied for more than 150 years. Classical solutions are all symmetric and involve both periodic waves in water of arbitrary
depth and solitary waves in water of finite depth. We first demonstrate
that there are in addition new pure gravity solitary waves in water of infinite
depth. We then include the effect of surface tension and present new types of
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non symmetric waves. The numerical methods used involve complex variable
techniques based on conformal mappings.
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