## Applied Mathematics Seminars

### Spring 2017

All seminars (unless otherwise stated) will take place on Tuesdays at 3.00pm in Room Archeology G6 (31-34 Gordon Square). See the map for further details. There will be tea afterwards in Mathematics Room 606 (25 Gordon Street). 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.

### 17 January 2017

#### Dr Max Jensen (Sussex University)

###### Abstract:

In this presentation I will discuss a semi-Lagrangian discretisation of the Monge-Ampère operator on P1 finite element spaces. The wide stencil of the scheme is designed to ensure uniform stability of numerical solutions. Monge-Ampère equations arise for example in the Inverse Reflector Problem where the geometry of a reflecting surface is reconstructed from the illumination pattern on a target screen and the characteristics of the light source.

Monge-Ampère type equations, along with Hamilton-Jacobi-Bellman type equations are two major classes of fully nonlinear second order partial differential equations (PDEs). From the PDE point of view, Monge-Ampère type equations are well understood.  On the other hand, from the numerical point of view, the situation is far from ideal. Very few numerical methods, which can reliably and efficiently approximate viscosity solutions of Monge-Ampère type PDEs on general convex domains. There are two main difficulties which contribute to the situation:

* Firstly, it is well known that the fully nonlinear structure and nonvariational concept of viscosity solutions of the PDEs prevent a direct formulation of any Galerkin-type numerical methods.

* Secondly, the Monge-Ampère operator is not an elliptic operator in generality, instead, it is only elliptic in the set of convex functions and the uniqueness of viscosity solutions only holds in that space.  This convexity constraint, imposed on the admissible space, causes a daunting challenge for constructing convergent numerical methods; it indeed screens out any trivial finite difference and finite element analysis because the set of convex finite element functions is not dense in the set of convex functions

The goal of our work is to develop a new approach for constructing convergent numerical methods for the Monge-Ampère Dirichlet problem, in particular, by focusing on overcoming the second difficulty caused by the convexity constraint. The crux of the approach is to first establish an equivalent (in the viscosity sense) Bellman formulation of the Monge-Ampère equation and then to design monotone numerical methods for the resulting Bellman equation on general triangular grids. An aim in the design of the numerical schemes was to make Howard's algorithm available, which is a globally superlinearly converging semi-smooth Newton solver as this allows us to robustly compute numerical approximations on very fine meshes of non-smooth viscosity solutions. An advantage of the rigorous convergence  analysis of the numerical solutions is the comparison principle for the Bellman operator, which extends to non-convex functions. We deviate from the established Barles-Souganidis framework in the treatment of the boundary conditions to address challenges arising from consistency and comparison. The proposed approach also bridges the gap between advances on numerical methods for these two classes of second order fully nonlinear PDEs.

The contents of the presentation is based on joint work with X. Feng from the University of Tennessee.

### 24 January 2017

#### Dr Angela Mihai (Cardiff University)

###### Abstract:

In some soft biological structures, such as brain, liver and fat tissues, strong experimental evidence suggests that the shear modulus increases significantly under increasing compressive strain, but not under tensile strain, while the apparent elastic modulus increases or remains almost constant when compressive strain increases. These tissues also exhibit a predominantly isotropic, incompressible behaviour. Our aim is to capture these seemingly contradictory mechanical behaviours, both qualitatively and quantitatively, within the framework of finite elasticity, by modelling a soft tissue as a homogeneous, isotropic, incompressible, hyperelastic material and comparing our results with available experimental data.  This is joint work with Prof. Alain Goriely (University of Oxford).

### 7 February 2017

#### Professor Andrew Soward (Newcastle Unviersity)

###### Abstract:

Greenspan and Howard (J. Fluid Mech., 1963) studied the linear spin-down of a rapidly rotating viscous fluid at small Ekman number E inside a container with rigid boundaries, following an instantaneous small change in container angular velocity. Outside the Ekman layers, thickness O(E^{1/2}), the mainstream is in almost rigid rotation (geostrophic) but spins down rapidly due to Ekman suction. Additionally, there are thickening quasi-geostrophic and very weak ageostrophic E^{1/3} shear layers adjacent to the cylindrical side-wall. Motivated by applications to isolated atmospheric structures (e.g.,~ tropical cyclones, tornadoes) without side and top boundaries, we study numerically and asymptotically a variant with stress-free side-wall and top boundaries, which leads to unexpected consequences. The mainstream no longer rotates rigidly, while the ageostrophic E^{1/3} shear layer, far from being passive, determines a spin-down rate dependent on ln E. It is linked to an E^{1/2} x E^{1/2} corner region, where the rigid base and the stress-free side-wall meet; a singularity that limits asymptotic progress.

### 21  February 2017

#### Dr Joel Daou (University of Manchester)

###### Abstract:

In 1953, the British physicist G.I. Taylor published an influential paper describing the enhancement of diffusion processes by a (shear) flow, a phenomenon later termed Taylor dispersion.  This has generated to date thousands of publications in various areas involving transport phenomena, none of which, surprisingly, in the field of combustion.

In 1940, the German chemist G. Damköhler postulated two hypotheses which have largely shaped current views on the propagation of premixed flames in turbulent flow fields.
According to  Damköhler’s first hypothesis, the large scales in the flow simply increase the flame surface area by wrinkling it, without affecting its local normal propagation speed.
According to  Damköhler’s second hypothesis, the small scales in the flow do not cause any significant flame wrinkling but do increase the normal propagation speed (and flame thickness). However, unlike the first hypothesis, the second one has received little support in the research literature, especially as far as analytical work is concerned.

We shall present analytical and numerical studies which are the first on Taylor dispersion in the context of combustion.  In particular, simple analytical formulas will be derived   which establish the link between Taylor dispersion and Damköhler’s second hypothesis. The findings   provide explanations, within simple analytical laminar-flow models, of the apparently contradictory results found in experimental and numerical studies on flame propagation in more complex (turbulent) flows. One such explanation is related to the so-called bending-effect of the turbulent flame speed, which is observed experimentally under high intensity turbulent flows. Another explanation is related to clarifying the dependence of the apparent Lewis number on the flow.

### 28 February 2017

#### Prof. Patrick Farrell (University of Oxford)

###### Abstract:

Computing the solutions $u$ of an equation $f(u, \lambda) = 0$ as the parameter $\lambda$ is varied is a central task in applied mathematics. In this talk I will present a new algorithm, deflated continuation, for this task.

Deflated continuation has three main advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only attempted to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to any existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available.

Among other problems, we will apply this to a famous singularly perturbed ODE, Carrier's problem. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold
bifurcations as the singular perturbation parameter tends to zero. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag (1999) is incorrect.

### 14 March 2017

#### Dr. Alden Waters (University College of London)

###### Abstract:

We prove time asymptotic observability estimates for the Schrodinger equation with a variety of potentials in L2(Ω) using the Sturm-Lioville theory. We prove existence of an observability constant under criterion related to the geometric control condition and describe its explicit properties. We give an explicit description of numerical models to analyze the properties well-known examples of potentials wells, including that of the harmonic oscillator.

### 21 March 2017

#### Dr Maria Korotyaeva (Institut FEMTO-ST)

###### Abstract:

We propose the resolvent method for calculating the horizontally polarized shear waves spectra in the 2D phononic crystal (PC) waveguides: the free PC plate and the PC plate sandwiched between two substrates. Since the propagator over a unit cell approximated by Fourier harmonics in one coordinate can have very large components, we introduce its resolvent as a numerically stable substitute.