ICMS

Boundary value problems for linear elliptic and integrable PDEs: theory and computation

May 28, 2012 - Jun 01, 2012

ICMS 15 South College Street

Organisers

Name	Institution
Bona, Jerry	University of Illinois at Chicago
Chandler-Wilde, Simon	University of Reading
Fornberg, Bengt	University of Colorado
Pelloni, Beatrice	University of Reading

 

IMPORTANT: EMAIL DISRUPTION We have been experiencing problems with email sent to all addresses in the icms.org.uk domain since the evening of Monday 21 May. If you have sent email to an icms.org.uk address since then it may well have been lost.   If you need to contact Audrey Brown, please use the following email address until the problem is rectified:   abrown16@staffmail.ed.ac.uk

 

The aim of this workshop is to bring together researchers who have contributed to recent developments in the theory and computation of boundary value problems for certain important partial differential equations, from varying perspectives and using disparate methods.

The meeting will focus on the following themes:

• boundary value problems for nonlinear PDEs, integrablibility formalism and PDE techniques
• boundary value problems for linear elliptic PDEs, Wiener-Hopf methods and high frequency scattering
• new numerical formalisms and comparison with existing numerical techniques, with an emphasis on high frequency wave problems.
  

Arrangements

Participation
Participation for this workshop is by invitation only.  

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one.

Venue
The workshop will be held at 15 South College Street, Edinburgh.  You may also find this map useful for the workshop.  All lectures will be held in the Newhaven Lecture Theatre.  To view this room and a list of the visual equipment available click here.  In addition, two blackboards have recently been installed.  The projector and one board may be used simultaneously.  

Wireless Access
The workshop venue, 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network.  For those without laptops, there will also be a couple of computers available for you to check your emails.

Travel
Information about travel to the UK and Edinburgh is available here.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies (volcanic ash disruptions) that may occur on your trip.

A taxi directly from the airport will cost approximately 15.00 to 20.00 GBP to the city centre for a one-way journey.     There is also a bus service direct from the airport to the city centre which will cost 3.50 single or 6.00 GBP return - the Airlink 100.  This is a frequent service (every 10 minutes during peak times) and will bring you close to Waverley Railway Station, only a short walk to the accommodation and the workshop venue (see map in 'Venue' section above). 

Lothian buses charge £1.40 for a single, £3.50 for a day ticket. Please note that the exact fare is required and no change is given.

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge. The second railway station is called Haymarket and is at the West End of the city centre.

Accommodation
If you asked ICMS to organise your accommodation, you will receive an email informing you whether you are staying at Masson House OR Kenneth MacKenzie Suite:
Map showing location of Masson House, Pollock Halls of Residence
Map showing location of Kenneth MacKenzie Suite 

Catering
Refreshments will be provided during the coffee breaks for the duration of the event.   A light lunch will be available in the Chapterhouse of 15 South College Street on Monday, Tuesday and Thursday.  Participants are free to make their own arrangements for lunch on Wednesday and the workshop closes at 12.30 on Friday.  There will be a workshop dinner in St Leonard's Hall at Pollock Halls of Residence, Holyrood Park Road, Edinburgh, on Thursday 31 May.

Financial Matters and Registration Fee
The workshop grant will cover the cost of bed and breakfast accommodation for all participants, refreshments during the coffee breaks for the duration of the workshop, 3 lunches, an informal wine reception on Monday evening and the workshop dinner at St Leonard's Hall on Thursday.

A registration fee of 50.00 GBP is payable by all.   If possible, we'd prefer that this fee is paid in advance of the meeting by completing the details on the credit/debit card payment form.   As email is not secure, please print the form and return it by fax using the fax number at the top of the form.   ICMS will take your payment and a receipt will be emailed to you.  Alternatively, we can accept sterling cheques, made payable to Heriot-Watt University but sent to Audrey Brown, ICMS, 15 South College Street, Edinburgh, EH8 9AA.  If you have any difficulty in paying in advance, it is possible for you to simply pay on arrival at Registration. 

If we have agreed to pay some of your travel costs the amount will be confirmed in your 'Final Information' email which you will receive a week prior to the workshop.  Reimbursement will take place after the event and will involve payment directly into your bank account.  At Registration you will be given an expenses claim form and this should be submitted to ICMS, with original receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from  the rest of the world, an IBAN and SWIFT/BIC code. We cannot reimburse any item without a signed claim form and original receipts.

 

Programme

Please continue to check this Programme as alterations may appear prior to the workshop.

Monday 28 May 2012

08.45-09.20	Registration & coffee
09.20-09.30	Welcome & introduction
09.30-10.30	Thanasis Fokas (University of Cambridge) An effective method for integrable nonlinear PDEs on the half-line
10.30-11.00	Coffee/Tea in Chapterhouse, Level 1
11.00-12:00	Euan Spence (University of Bath) New and old methods for solving linear elliptic PDEs numerically
12:00-13:00	Natasha Flyer (National Center for Atmospheric Research) Improving numerical accuracy in the presence of corner singularities
13:00-14.00	Lunch/Discussions (lunch provided in the Chapterhouse, Level 1)
14.00-15.00	Shu-Ming Sun (Virginia Polytechnic Institute and State University) Nonhomogenous boundary value problems of the Schrodinger equations
15.00-16.00	Alex Himonas (University of Notre Dame) The initial value problem for a class of integrable equations
16:00-16.30	Coffee/Tea in Chapterhouse, Level 1
16.30-17.30	Gino Biondini (State University of New York at Buffalo) On the Maxwell-Bloch equations with non-zero boundary conditions
17.30 – 19.00	Informal Wine Reception in the Atrium, Level 3

 

Tuesday 29 May 2012

09.00-10:00	Ralf Hiptmair (ETH Zurich) Novel boundary element methods for scattering at composite objects
10.00-11.00	Anthony Ashton (University of Cambridge) On the rigorous foundations of the Fokas transform method
11.00-11.30	Coffee/Tea in Chapterhouse, Level 1
11.30-12.30	Daan Huybrechs (KU Leuven) Numerical-asymptotic methods for solving the Helmholtz equation in the high-frequency regime
12.30-13.30	Sheila Smitheman (University of Nottingham) TBA
12.30-14.00	Lunch/Discussions (lunch provided in the Chapterhouse, Level 1)
14.00-15.00	Ivan Graham (University of Bath) TBA
15.00-16.00	Ohannes Karakashian (University of Tennessee) Conservative discontinuous Galerkin methods for the generalized Korteweg-de Vries equation
16.00-16.30	Coffee/Tea in Chapterhouse, Level 1
16.30-17.30	Elizabeth Its (Indiana University Purdue University Indianapolis) Riemann-Hilbert approach to scattering problems in elastic media

 

Wednesday 30 May 2012

09.00-10:00	Bengt Fornberg (University of Colorado) The Fokas transform approach for Laplace- and Helmholtz-type equations: a spectrally accurate implementation of the Dirichlet-to-Neumann map problem on a polygonal domain
10:00-11.00	Stephen Langdon (University of Reading) A high frequency boundary element method for scattering by a class of nonconvex obstacles
11.00-11.30	Coffee/Tea in Chapterhouse, Level 1
11.30-12.30	Jean-Claude Saut (Université Paris Sud) Diffusive effects of boundary conditions for nonlinear dispersive equations
12.30-13:30	Lunch break (no lunch provided)
13.30-15.00	Discussion Session – then free afternoon

 

Thursday 31 May 2012

09.00-10:00	Darren Crowdy (Imperial College London) Mixed boundary value problems in the study of superhydrophobic surfaces
10:00-11:00	David Hewett (University of Reading) High frequency scattering by screens: boundary integral equation formulations
11.00-11.30	Coffee/Tea in Chapterhouse, Level 1
11.30-12.30	Dimitra Antonopoulou (University of Crete) The outer parabolic Stefan problem for Ostwald ripening with kinetic undercooling and inhomogeneous driving force
12.30-14.00	Lunch/Discussions (lunch provided in the Chapterhouse, Level 1)
14.00-15.00	Dionyssis Mantzavinos (University of Cambridge) Initial-boundary value problems for linear and nonlinear evolution PDEs
15.00-16.00	Thomas Trogdon (University of Washington) Numerical inverse scattering: uniform approximation of integrable PDEs
16.00-16.30	Coffee/Tea in Chapterhouse, Level 1
16.30-17.30	David Smith (University of Crete) Well-posedness of linear initial-boundary value problems
19.00	Workshop dinner at St Leonard’s Hall, Pollock Halls of Residence, Edinburgh

 

Friday 1 June 2012

09.00-10.00	Jonatan Lenells (Baylor University) The Fokas method for equations with 3x3 Lax pairs
10:00-11.00	Hongqiu Chen (University of Memphis) Stability of solitary-wave solutions for some dispersive systems of equations
11:00-11.30	Coffee/Tea in Chapterhouse, Level 1
11.30-12.30	Jerry Bona (University of Illinois at Chicago) Boundary-value problems arising in water wave models
12.30	Closing

 

Presentations:

Presentation Details
Antonopoulou, Dimitra
The outer parabolic Stefan problem for Ostwald ripening with kinetic undercooling and inhomogeneous driving force
View Abstract Hide Abstract
Ostwald ripening is the coarsening phenomenon caused by the diffusion and solidification process which occurs in the last stage of a first-order phase transformation. The force that drives the system towards equilibrium is the gradient of the chemical potential that, according to the Gibbs-Thomson condition, on the interface, is proportional to its mean curvature. A quantitative description of Ostwald ripening has been developed by the Lifschitz-Slyozov-Wagner (LSW) theory. We extend the work of Niethammer which deals with kinetic undercooling in the quasi-static case to the parabolic setting with temporally inhomogeneous driving forces on the solid-liquid interfaces. By means of a priori estimates, local and global existence results for the parabolic Stefan problem, we derive a first order approximation for the dynamical equations for the heat distribution and particle radii and then prove the convergence to a limiting description using a mean-field equation.
Ashton, Anthony
On the rigorous foundations of the Fokas transform method
View Abstract Hide Abstract
We will discuss some of the rigorous aspects of the Fokas transform method, focusing on elliptic boundary value problems on convex domains. The central object in the Fokas method is the global relation: an integral equation in spectral space that couples the known boundary data with the unknown boundary values. Using this approach we explore several rigorous problems (a) existence of a solutions to the global relation; (b) local regularity of the generalised Dirichlet-Neumann map; (c) distributional boundary data on convex polygons.
Biondini, Gino
On the Maxwell-Bloch equations with non-zero boundary conditions
View Abstract Hide Abstract
The initial-boundary value problem for the Maxwell-Bloch equations with non-zero fields at infinity is considered. The inverse scattering transform is formulated and is used to investigate the behaviour of the solutions.
Bona, Jerry
Boundary-value problems arising in water wave models
View Abstract Hide Abstract
The lecture will survey some recent results about boundary-value problems arising in water wave theory.
Chen, Hongqiu
Stability of solitary-wave solutions for some dispersive systems of equations
View Abstract Hide Abstract
In the talk, a class of systems of two non-linear dispersive equations is introduced. Up to three solitary-wave solutions are found explicitly, their orbital stability is analyzed. The result is more subtle and fascinating than they appear.
Crowdy, Darren
Mixed boundary value problems in the study of superhydrophobic surfaces
View Abstract Hide Abstract
To follow
Flyer, Natasha
Improving numerical accuracy in the presence of corner singularities
View Abstract Hide Abstract
Rarely is initial and boundary data compatible with the evolutionary PDE in the corners of the spatial-temporal domain. These incompatibles introduce errors that destroy the exponential accuracy of spectral methods. For evolutionary equations, these errors can be localized in time, in both space and time or in neither, depending on the properties of the PDE. We will examine three numerical techniques to improve accuracy in the face of these ever-present corner singularities.
Fokas, Thanasis
An effective method for integrable nonlinear PDEs on the half-line
View Abstract Hide Abstract
A new method for analyzing boundary value problems (BVPs) extending ideas of the inverse scattering transform method, was introduced in [1] and further developed by several researchers. Recent unexpected developments regarding the implementation of this method to evolution PDEs in the half-line will be reviewed, which show that: (i)For linearizable BVPs the new method is as effective as the usual inverse scattering transform method; this includes PDEs with a third order derivative, for which the alternative approach of extension from the half-line to the full line, fails. (ii) For general (non-linearizable) BVPs, it yields the solution via a matrix Riemann– Hilbert problem, whose jump matrix can be characterized via a well defined nonlinear equation, in terms of the given initial and boundary conditions. (iii) For general BVPs, with either decaying, or t-periodic boundary conditions, it yields effective long time asymptotic formulae. References [1] A.S. Fokas, A Unified Transform Method for Solving Linear and Certain Nonlinear PDEs, Proc. R. Soc. Lond. A 453, 1411–1443 (1997).
Fornberg, Bengt
The Fokas transform approach for Laplace - and Helmholtz - type equations: a spectrally accurate implementation of the Dirichlet-to-Neumann map problem on a polygonal domain
View Abstract Hide Abstract
Both boundary element and boundary integral methods can provide numerical solutions to the Dirichlet-to-Neumann map problem for many linear elliptic PDEs, without the need for discretizations throughout the domain interiors. The Fokas transform provides a fundamentally different approach for this, with several noteworthy features. When utilizing Legendre expansions along the sides of a polygonal domain, all the integrals that arise become available in simple closed form. As a result, the Dirichlet-to-Neumann map problem requires only the solution of a small overdetermined linear system. Monitoring singularities of this system provides an easy approach for calculating eigenvalues of the Helmholtz operator. Still another strength with the new approach is that one readily can incorporate corrections for frequently arising corner singularities.
Hewett, David
High frequency scattering by screens: boundary integral equation formulations
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We consider the scattering of time-harmonic waves by a flat screen in two or three dimensions. We consider first the acoustic case with Dirichlet (sound soft) or Neumann (sound hard) boundary conditions. For both boundary conditions we prove coercivity of the relevant boundary integral formulations, and provide new frequency-explicit estimates of the associated coercivity and continuity constants. We also discuss the extension to the electromagnetic case, and the implication of these results for the convergence of Galerkin boundary element methods for high frequency screen problems.
Himonas, Alex
The initial value problem for a class of integrable equations
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Focusing on the Korteweg-de Vries (KdV) and the Camassa-Holm (CH) equations we shall present a few ideas and techniques for studying the Cauchy problem for such equations when the initial data belong in spaces of analytic functions. Also, we shall provide an overview of well-posedness results for these equations when the initial data belong to rougher spaces.
Hiptmair, Ralf
Novel boundary element methods for scattering at composite objects
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Classical boundary integral formulations for multi-domain transmission problems for time-harmonic acoustic or electromagnetic waves involve first-kind equations that lead to ill-conditioned linear systems when discretized by means of boundary elements. Recently, two remedies have been explored: (i) So-called multi-trace formulations that employ a domain decomposition idea to obtain integral equations amenable to Calderon preconditioning. (ii) Intrinsically well-conditioned 2nd-kind boundary integral equations.
Huybrechs, Daan
Numerical-asymptotic methods for solving the Helmholtz equation in the high-frequency regime
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The Helmholtz equation is perhaps the `simplest difficult equation'. Though simple in form, large values of the wavenumber produce highly oscillatory solutions. As a result, standard fast numerical methods for solving PDE's quickly loose their appeal in various ways. The most insightful analysis of the Helmholtz equation in the high-frequency regime comes from asymptotic analysis. It is our purpose to illustrate that such analysis is also a good starting point for numerical methods. One important design criterion for a numerical method in the high-frequency regime is to retain two favourable properties of asymptotic expansions: improving accuracy with increasing frequency, whilst retaining a bounded computational cost. In particular, we show that the accuracy of the widely used Geometrical Optics approximation, as well as some elements of the Geometric Theory of Diffraction, can be substantially improved with a small numerical computation. Asymptotic expansions are never explicitly constructed: it is sufficient to know that they exist. One advantage is that these numerical methods can extend the applicability range of asymptotic methods to include the so-called mid-frequency regime. A disadvantage is that they apply only in cases where the asymptotic analysis applies. This may entail stringent convexity requirements, for example, in at least part of a configuration of scattering obstacles.
Its, Elizabeth
Riemann-Hilbert approach to scattering problems in elastic media
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We are developing Riemann-Hilbert (RH) approach to scattering problems in elastic media. The approach is based on a version of RH method introduced in ninties by A. Fokas for studying boundary problems for linear and integrable nonlinear PDEs. The suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations obtained from this vector Lax pair are applied to Rayleigh wave propagation in an elastic quarter space and half space. This reduces the problem to the analysis of certain underdetermined matrix RH problem on a torus. We showed that the problem can be in fact re-formulated as a well-posed RH problem with a shift. Some results of the described analysis will be discussed. This is a joint project with Alexander Its and Julius Kaplunov.
Karakashian, Ohannes
Conservative discontinuous Galerkin methods for the generalized Korteweg-de Vries equation
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We construct, analyze and numerically validate a class of conservative, discontinuous Galerkin schemes for the Generalized Korteweg-de Vries equation. Up to round-off error, these schemes preserve discrete versions of the first two invariants (the integral of the solution, usually identified with the mass, and the $L^2$--norm) of the continuous solution. Numerical evidence is provided indicating that these conservation properties impart the approximations with beneficial attributes, such as more faithful reproduction of the amplitude and phase of travelling-wave solutions. The numerical simulations also indicate that the discretization errors grow only linearly as a function of time.
Langdon, Stephen
A high frequency boundary element method for scattering by a class of nonconvex obstacles
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In this talk, we discuss a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. We believe this to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle.
Lenells, Jonatan
The Fokas method for equations with 3x3 Lax pairs
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I will describe how the Fokas method can be generalized to the case of initial-boundary value problems for integrable evolution equations with 3x3 Lax pairs.
Mantzavinos, Dionyssis
Initial-boundary value problems for linear and nonlinear evolution PDEs
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There exist several powerful techniques for the solution of boundary value problems regarding linear PDEs, such as integral transforms (Fourier, Laplace and variations), Green's functions and the method of images; however, much less has been developed as far as nonlinear PDEs are concerned. Moreover, even in the case of linear problems one often comes across complicated boundary conditions which are not separable and thus cannot be treated by the standard classical techniques. This talk will be on the application of the Fokas method (introduced in 1997) to certain linear initial-boundary value problems with various types of boundary conditions which are not necessarily separable, as well as on the extension of the method to nonlinear evolution PDEs in two spatial dimensions using Lax pairs and an appropriate d-bar formalism.
Saut, Jean-Claude
Diffusive effects of boundary conditions for nonlinear dispersive equations
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We will present some results on the diffusive effects induced by boundary conditions on nonlinear dispersive equations.
Smith, David
Well-posedness of linear initial-boundary value problems
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We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also discuss the spectral properties of associated ordinary differential operators.
Smitheman, Sheila
Spectral methods for elliptic equations in regular and convex irregular polygons
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For linear PDEs, the new transform method introduced by Fokas in 1997 for solving BVPs can be considered as the analogue of the Green's function approach in the Fourier instead of the physical plane. It employs a global relation, also formulated in the complex plane, which couples the solution and its normal derivative on the boundary. The discretization of this relation has formed the basis of various research groups' efforts to compute the "Dirichlet to Neumann map" for the Laplace, modified Helmholtz and Helmholtz equations. In the first part of my talk, I will discuss relationships between the work of these groups, and I will outline the relative strengths and weaknesses of each approach. I will then explain how we are currently combining aspects of these existing approaches, together with some crucial new insights, to derive and implement an elegant and essentially analytic scheme for approximating the Dirichlet boundary data (rather than, as is usually the case, its tangential derivative) from given Robin boundary data for the three above basic linear, elliptic PDEs formulated in the interior of a regular polygon. Our approach appears to yield both spectral accuracy and a well conditioned system matrix whose structure reflects the rotational symmetry of the domain, allowing its inverse to be computed efficiently. (Joint work with A S Fokas, DAMTP, Cambridge)
Spence, Euan
New and old methods for solving linear elliptic PDEs numerically
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This talk will survey some different methods for solving linear elliptic PDEs numerically. The main goal of the talk is to link the ideas arising from the area of integrable systems (in the form of the Fokas method) to well-established ideas that numerical analysts will be familiar with.
Sun, Shu-Ming
Nonhomogenous boundary value problems of the Schrodinger equations
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The talk will discuss the initial- and boundary-value problems (IBVP) of the Schrodinger equation posed in a quarter plane and on a bounded interval with non-homogeneous boundary conditions. It will be shown that the IBVP is locally and globally well-posed in certain Banach spaces. The local well-poseness results are sharp, while the partial global well-posedness is obtained. The ideas used for the local well-posedness is similar to our early results on the KdV equation. Moreover, the extension to higher dimensional cases for the spatial variables will be mentioned. (This is a joint work with J. Bona and B. Zhang)
Trogdon, Thomas
Numerical inverse scattering: uniform approximation of integrable PDEs
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Riemann-Hilbert problems arise in many different applications, ranging from integrable systems and special function theory to card-shuffling and random matrix theory. A recent collocation method allows for the numerical solution of Riemann-Hilbert problems. In this talk I will lay out how to use this method to, in effect, solve integrable equations on the whole line for arbitrary spectral data. Uniformity statements about the approximation can be proved. Additionally, this can be used to examine the validity of known asymptotic formulas.

Participants

Name	Institution
Antonopoulou, Dimitra	University of Crete
Ashton, Anthony	University of Cambridge
Biondini, Gino	State University of New York at Buffalo
Bona, Jerry	University of Illinois at Chicago
Chandler-Wilde, Simon	University of Reading
Chen, Hongqiu	University of Memphis
Clarkson, Peter	University of Kent
Crowdy, Darren	Imperial College London
Flyer, Natasha	National Center for Atmospheric Research
Fokas, Thanasis	University of Cambridge
Fornberg, Bengt	University of Colorado
Graham, Ivan	University of Bath
Hewett, David	University of Reading
Himonas, Alex	University of Notre Dame
Hiptmair, Ralf	ETH Zurich
Huybrechs, Daan	KU Leuven
Its, Alexander	Indiana University-Purdue University Indianapolis
Its, Elizabeth	Indiana University Purdue University Indianapolis
Karakashian, Ohannes	University of Tennessee
Langdon, Stephen	University of Reading
Lenells, Jonatan	Baylor University
Mansfield, Elizabeth	University of Kent
Mantzavinos, Dionyssis	University of Cambridge
Moiola, Andrea	University of Reading
Pelloni, Beatrice	University of Reading
Prinari, Barbara	University of Colorado Colorado Springs & University of Salento
Saut, Jean-Claude	Université Paris Sud
Shargorodsky, Eugene	King's College London
Smith, David	University of Crete
Smitheman, Sheila	University of Nottingham
Spence, Euan	University of Bath
Sun, Shu-Ming	Virginia Polytechnic Institute and State University
Trogdon, Thomas	University of Washington