## Applied and Computational Mathematics Seminars

### Programme

These seminars take place on Tuesdays, in Room M/2.06, Senghennydd Road, Cardiff from 3pm, unless otherwise stated.

When a seminar is not scheduled there is a collaborative workshop with other groups within the College of Physical Sciences & Engineering or a SIAM Chapter Meeting. Further details can be found on the School Diary.

For more information or if you wish to give a talk, please contact the programme organiser Dr Angela Mihai.

##### 7 October 2014

**Speaker:** Daniel Lesnic (University of Leeds).

**Title:** Determination of a force function in the wave equation.

**Abstract:** The determination of an unknown space- or time-dependent force function acting on a vibrating structure from boundary, interior or integral observations are investigated. Sufficient conditions for the uniqueness of solution are provided. These linear inverse force problems are ill-posed since small errors in the input data cause large errors in the output force solution. Consequently, when the input data is contaminated with noise, we use regularization methods, e.g. Tikhonov's regularization, or conjugate gradient method, in order to obtain a stable solution. Numerical results will be presented and discussed.

##### 14 October 2014

**Speaker:** Timo Betcke (University College London).

**Title:** A spectral view on convolution quadrature methods for the wave equation.

**Abstract:** Convolution Quadrature (CQ) methods are Laplace transform type methods for the solution of time-domain wave problems. They are particularly popular for the solution of exterior time-domain wave scattering problems using boundary integral equation formulations in the Laplace domain. In this talk we will take a spectral view on CQ methods and discuss the connection between scattering poles of the solution operator, the underlying time-stepping scheme and convergence of CQ methods. The presented numerical examples are computed with BEM++, an open-source boundary element library developed at UCL. We will give a brief overview of BEM++ and demonstrate its functionality for solving boundary integral equations.

##### 28 October 2014

**Speaker:** John Pryce (Cardiff School of Mathematics).

**Title:** The Forthcoming IEEE 1788 Standard for Interval Arithmetic.

**Abstract:** Interval arithmetic (IA) is the most used way of producing rigorously proven results in problems of continuous mathematics, usually in the form of real intervals that (even in presence of rounding error) are guaranteed to enclose a value of interest, such as a solution of a differential equation at some point. The basics of IA are generally agreed e.g., to add two intervals xx, yy, find an interval containing all x + y for x in xx and y in yy.

Many versions of IA theory exist, individually consistent but mutually incompatible. They differ especially in how to handle operations not everywhere defined on their inputs, such as division by an interval containing zero. In this situation a standard is called for, which not all will love but which is usable and practical in most IA applications.

The IEEE working group P1788, begun in 2008, has produced a draft standard for interval arithmetic, currently undergoing the IEEE approval process. The talk will concentrate on aspects of its architecture, especially:

- the levels structure, with a mathematical, a datum and an implementation level;

- the decoration system, which notes when a library operation is applied to input where it is discontinuous or undefined.

Time permitting, I may outline the P1788 flavor concept, by which implementations based on other versions of IA theory may be included into the standard in a consistent way.

##### 11 November 2014

**Speaker:** Natalia Kopteva (University of Limerick, Ireland).

**Title:** Maximum norm a posteriori error estimates for parabolic partial differential equations.

**Abstract:** Solutions of partial differential equations frequently exhibit corner singularities and/or sharp boundary and interior layers. To obtain reliable numerical approximations of such solutions in an efficient way, one may want to use meshes that are adapted to solution singularities. Such meshes can be constructed using a priori information on the solutions, however it is rarely available in real-life applications. Therefore the best hope for the future seems to be offered by the automated mesh construction by adaptive techniques. This approach requires no initial asymptotic understanding of the nature of the solutions and the solution singularity locations.

Reliable adaptive algorithms are based on a posteriori error estimates, i.e. estimates of the error in terms of values obtained in the computation process: computed solution and current mesh. Such a posteriori error estimates for parabolic partial differential equations will be the subject of this talk. For classical and singularly perturbed semilinear parabolic equations, we give computable a posteriori error estimates in the maximum norm, which, in the singularly perturbed regime, hold uniformly in the small perturbation parameter. The parabolic equations are discretized in time using the backward Euler, Crank-Nicolson and discontinuous Galerkin methods. Both semidiscrete (no spatial discretization) and fully discrete cases will be considered. The analysis invokes certain bounds for the Green's function of the parabolic operator. When dealing with the full discretizations, we also employ the elliptic reconstruction technique.

Although parts of our analysis are quite technical, it will be demonstrated (using a first-order ODE example as a trivial case of a parabolic PDE) that some main ideas are quite elementary.

##### 18 November 2014

**Speaker:** Robert Style (University of Oxford).

**Title:** The mechanics of soft solids - breaking classical laws.

**Abstract:** Soft solids make up the bulk of biological material, and are increasingly being used for new technology like wearable electronics, and soft robotics. However, despite their importance, experiments show that many classical laws fail to describe them. For example, I will show how classical theories of wetting, composite mechanics and contact mechanics significantly break down at a critical `elastocapillary' lengthscale -- because of solid surface tension. Furthermore, I will show how these phenomena highlight the existence of a swathe of new, small-scale behaviour in soft materials.

Co-Host: Dr. Maurice Blount.

##### 25 November 2014

**Speaker:** Xuesong Wu (Imperial College London).

**Title:** Nonlinear development of disturbances in transitional and turbulent free shear flows.

**Abstract:** Free shear flows, such as mixing layers, jets and wakes, are inviscidly unstable due to their inflectional velocity profiles. Instability modes, which are usually excited by external perturbations, amplify on the shear floow, leading to vortex roll-up and randomization in the nonlinear stage. Interestingly, in turbulent state free shear flows exhibit a high degree of order, characterised by the prevalent presence of so-called coherent structures, the most striking of which are Brown-Roshko rollers .

Both instability waves and coherent structures are known to be dynamically significant for entrainment and mixing, noise generation as well as for turbulence modelling. In this talk, I will present a nonlinear theory to describe first the development of instability modes on laminar free shear layers. The theory predicts vortex roll-up and randomisation through a generalized side-band instability mechanism. The theory will then be modified to describe formation and evolution of Brown-Roshko rollers on turbulent mixing layers.

Co-Host: Dr. Chris Davies

##### 2 December 2014

**Speaker:** Matthias Heil (University of Manchester).

**Title:** Wrinkly fingers: fluid-structure interaction in elastic-walled Hele-Shaw cells.

**Abstract:** Viscous fingering in Hele-Shaw cells is a classical and widely studied fluid-mechanical instability: When air is injected into the narrow, liquid-filled gap between parallel rigid plates, the axisymmetrically expanding air-liquid interface tends to be unstable to non-axisymmetric

perturbations when the capillary number -- the ratio of (destabilising) viscous to (stabilising) capillary forces acting at the air-liquid interface -- becomes sufficiently large. The introduction of wall elasticity (via the replacement of one of the bounding plates by an elastic membrane) can weaken or even suppress the fingering instability, but it also makes the system susceptible to additional solid-mechanical instabilities.

We show that in elastic-walled Hele-Shaw cells that are bounded by sufficiently thin elastic sheets, the (fluid-based) viscous fingering instability can arise concurrently with a (solid-based) wrinkling instability. We study the interaction between these distinct instabilities, using a theoretical model that couples the depth-averaged lubrication equations for the fluid flow to the Föppl--von Kármán equations, which describe the deformation of the thin elastic sheet.

We employ a linear stability analysis to determine the growth rate of non-axisymmetric perturbations to the axisymmetrically expanding bubble,

and perform direct numerical simulations to study the nonlinear interactions between the instabilities. We show that the system's behaviour may be characterised by a non-dimensional parameter that indicates the strength of the fluid-structure interaction. For small [large] values of this parameter the system's behaviour is dominated by viscous fingering [wrinkling], with strong interactions between the two instabilities arising in an intermediate regime. [Joint work with Draga Pihler-Puzovic and Anne Juel].

Co-Host: Dr. Chris Davies

##### 3 February 2015

**Speaker:** Helen Wilson (University College London).

**Title:** Instabilities in viscoelastic fluids.

**Abstract:** Non-Newtonian and viscoelastic fluids show many fascinating properties. One of the most surprising (and irritating, for those who process them in a manufacturing context) is their susceptibility to instabilities at flow rates where an "equivalent" Newtonian fluid would flow stably. I will use linear stability theory, and some asymptotic expansions, to discuss two distinct instability mechanisms which are unique to viscoelastic fluids.

Co-Host: Professor Tim Phillips.

##### 10 February 2015

**Speaker:** Dmitri Tseluiko (University of Loughborough).

**Title:** Wave dynamics on a liquid film sheared by a turbulent gas

**Abstract:** The dynamics of a thin laminar liquid film flowing under gravity down the lower wall of an inclined channel when turbulent gas flows above the film will be discussed. The solution of the full system of equations describing the gas-liquid flow faces serious technical difficulties. However, a number of assumptions allow isolating the gas problem and solving it independently by treating the interface as a solid wall. This permits finding the perturbations to normal and tangential stresses at the interface imposed by the turbulent gas in closed form. Then the liquid film flow under the influence of these perturbations can be analysed by deriving and analysing a hierarchy of model equations describing the dynamics of the interface, i.e. boundary-layer equations, a long-wave model and a weakly nonlinear model, which turns out to be the Kuramoto-Sivashinsky equation with an additional term due to the presence of the turbulent gas. Also, by combining the long-wave approximation with a weighted-residual technique, an integral-boundary-layer approximation that is valid for moderately large values of the Reynolds number can be obtained. This model is then used for a systematic investigation of the flooding phenomenon observed in various experiments: as the gas flow rate is increased, the initially downward-falling film starts to travel upwards while just before the wave reversal the amplitude of the waves grows rapidly.

Co-Host: Dr Nikos Savva.

##### 17 February 2015

**Speaker:** David Needham (University of Birmingham).

**Title:** The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data.

**Abstract:** This talk addresses the Cauchy Problem for a Non-Lipschitz Semi-Linear parabolic PDE with trivial initial data. The question of uniqueness is considered, in particular in relation to the existence of classes of spatially inhomogeneous solutions, and their relation to maximal and minimal solutions.

**Co-Host**: Professor Tim Phillips.

##### 17 March 2015

**Speaker:** Paul Milewski (University of Bath).

**Title:** Modelling and computation of pilot wave-bouncing droplet dynamics in a Faraday problem.

**Abstract:** Recent experiments by two groups, Yves Couder (Paris) and John Bush (MIT) have shown experimentally that droplets will bounce on the surface of a vertically vibrated bath (instead of coalescing with it), generating a damped Faraday wavefield at every bounce. As the forcing is increased, a pitchfork symmetry breaking bifurcation leads to a "walking" state whereby the bouncing droplet is guided by the self-generated wavefield: the droplet s pilot wave. Once this state is achieved a large array of interesting dynamics ensues with surprising analogies to quantum mechanical behaviour. The system appears to show that probabilistic quantum behaviour can arise from a physical deterministic system. We present a coupled particle-fluid model that can can be used simulate the fascinating dynamics of this problem. This is joint work with John Bush, Andre Nachbin (IMPA) and Carlos Galeano-Rios (IMPA).

Co-Host: Dr Nikos Savva.