Internal waves in lakes and oceans are ubiquitous whenever a density stratification is present. These waves are relatively slow moving, can be large in extent and have long time scales. As these waves are so common, it is suspected that they play a role in recirculating nutrients throughout the water column. The various factors contributing to this recirculation are commonly referred to as the nutrient circulation system. This thesis analyses three potential mechanisms of internal wave forcing of the nutrient circulation system over a range of length scales. Namely, we discuss internal wave shear induced sediment resuspension, non-Newtonian fluid mud vortex dynamics and internal wave forced lake bottom seepage. We believe that these demonstrate the significant effect that internal waves can have on distributing nutrients throughout the water column. In conjunction, these mechanisms have the potential to be the dominant source of nutrient circulation in certain regions of lakes and oceans.

Goal of this thesis is to study four problems. In chapters 3-5, we consider scalar conser- vation law in one space dimension with strictly convex flux. First problem is to know the profile of the entropy solution. In spite of the fact that, this was studied extensively in last several decades, the complete profile of the entropy solution is not well understood. Second problem is the exact controllability. This was studied for Burgers equation and some partial results are obtained for large time. It was a challenging problem to know the controllability for all time and also for general convex flux. In a seminal paper [25], Dafermos introduces the characteristic curves and obtain some qualitative properties of a solution of a convex conservation law. In this thesis, we further study the finer properties of these characteristic curves. Here we solve these two problems in complete generality. In view of the explicit formulas of Lax - Oleinik [31], Joseph - Gowda [40], target func- tions must satisfy some necessary conditions. In this thesis we prove that these are also sufficient. Method of the proof depends highly on the characteristic methods and explicit formula given by Lax - Oleinik and the proof is constructive. Third problem is to solve the optimal controllability problem. In chapter 5 we derive a method to obtain a solution of an optimal control problem for the scalar conservation laws with convex flux. By using the method of descent, this type of problem was considered by Castro-Palacios-Zuazua in [23] for the Burgers equation. Our approach is simple and based on the explicit formulas of Hopf and Lax-Olenik. Last but not the least is about the problem of total variation bound for solution of scalar conservation laws with discontinuous flux. For the scalar con- servation laws with discontinuous flux, an infinite family (A, B)-interface entropies are introduced and each one of them has been shown to form an L1 -contraction semigroup (see, [8]). One of the main unsettled questions concerning conservation law with discon- tinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [16]. In the chapter 6, we discuss this particular issue in detail and produce a counter example to show that the solution, in general, has unbounded total variation near the interface. In fact this example illustrates that smallness of BV norm of the initial data is immaterial. We hereby settled the question of determining for which of the aforementioned (A, B) pairs, the solution will have bounded total variation in case of strictly convex fluxes.

Internal waves in lakes and oceans are ubiquitous whenever a density stratification is present. These waves are relatively slow moving, can be large in extent and have long time scales. As these waves are so common, it is suspected that they play a role in recirculating nutrients throughout the water column. The various factors contributing to this recirculation are commonly referred to as the nutrient circulation system. This thesis analyses three potential mechanisms of internal wave forcing of the nutrient circulation system over a range of length scales. Namely, we discuss internal wave shear induced sediment resuspension, non-Newtonian fluid mud vortex dynamics and internal wave forced lake bottom seepage. We believe that these demonstrate the significant effect that internal waves can have on distributing nutrients throughout the water column. In conjunction, these mechanisms have the potential to be the dominant source of nutrient circulation in certain regions of lakes and oceans.

We consider reaction-diffusion systems of two variables with Neumann boundary conditions on a finite interval with diffusion rates of different orders. Solutions of these systems can exhibit a variety of patterns and behaviours; one common type is called a mesa pattern; these are solutions that in the spatial domain exhibit highly localized interfaces connected by almost constant regions. The main focus of this thesis is to examine three different mechanisms by which the mesa patterns become unstable. These patterns can become unstable due to the effect of the heterogeneity of the domain, through an oscillatory instability, or through a coarsening effect from the exponentially small interaction with the boundary. We compute instability thresholds such that, as the larger diffusion coefficient is increased past this threshold, the mesa pattern transitions from stable to unstable. As well, the dynamics of the interfaces making up these mesa patterns are determined. This allows us to describe the mechanism leading up to the instabilities as well as what occurs past the instability threshold. For the oscillatory solutions, we determine the amplitude of the oscillations. For the coarsening behaviour, we determine the motion of the interfaces away from the steady state. These calculations are accomplished by using the methods of formal asymptotics and are verified by comparison with numerical computations. Excellent agreement between the asymptotic and the numerical results is found.

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This thesis presents new numerical methods for solving differential equations with periodicity. Spectral methods for solving linear and nonlinear ODEs, linear ODE eigenvalue problems and linear time-dependent PDEs on a periodic interval are reviewed, and a novel approach for computing multiplication matrices is presented. Choreographies, periodic solutions of the n-body problem that share a common orbit, are computed for the first time to high accuracy using an algorithm based on approximation by trigonometric polynomials and optimization techniques with exact gradient and exact Hessian matrix. New choreographies in spaces of constant curvature are found. Exponential integrators for solving periodic semilinear stiff PDEs in 1D, 2D and 3D periodic domains are reviewed, and 30 exponential integrators are compared on 11 PDEs. It is shown that the complicated fifth-, sixth- and seventh-order methods do not really outperform one of the simplest exponential integrators, the fourth-order ETDRK4 scheme of Cox and Matthews. Finally, algorithms for solving semilinear stiff PDEs on the sphere with spectral accuracy in space and fourth-order accuracy in time are proposed. These are based on a new variant of the double Fourier sphere method in coefficient space and standard implicit-explicit time-stepping schemes. A comparison is made against exponential integrators and it is found that implicit-explicit schemes perform better. The algorithms described in each chapter of this thesis have been implemented in MATLAB and made available as part of Chebfun.

Shocks form in the solutions of first-order nonlinear hyperbolic PDEs with constant co-efficients. Where solitary waves arise in the solutions of first-order nonlinear hyperbolic PDEs with variable coefficients, those solitary waves occur due to the coupling of nonlinearity and dispersive effects that comes from the medium’s heterogeneity. In this thesis, we study a fluid that propagates in a narrow pipe with periodically-varying cross-sectional area described by a system of first-order nonlinear hyperbolic PDEs. Multiple-scale perturbation theory is applied to derive homogenized effective equations, which take the form of a constant-coefficient system including higher-order dispersive terms. We investigate the behavior of the solution by deriving the linear dispersion relation of the homogenized system. The homogenized equations are solved using a psuedospectral discretization in space and explicit Runge-Kutta method in time. Lastly, we develop a Riemann solver in Clawpack to solve the variable coefficients system and compare the obtained solution with the homogenized equations solution.

No / This paper presents a methodology for efficient shape parametrisation for automatic design optimisation using a partial differential equation (PDE) formulation. It is shown how the choice of an elliptic PDE enables one to define and parametrise geometries corresponding to complex shapes. By using the PDE formulation it is shown how the shape definition and parametrisation can be based on a boundary value approach by which complex shapes can be created and parametrised based on the shape information at the boundaries or the character lines defining the shape. Furthermore, this approach to shape definition allows complex shapes to be parametrised intuitively using a very small set of design parameters.

In the setting of Riemannian manifolds with nonnegative Ricci curvature, we provide geometric inequalities as consequences of the Monotonicity Formulas holding along the flow of the level sets of the p-capacitary potential. The work is divided into three parts. (1) In the first part, we describe the asymptotic behaviour of the p-capactitary potential in a natural class of Riemannian manifolds. (2) The second part is devoted to the proof of our Monotonicity-Rigidity Theorems. (3) In the last part, we apply the Monotonicity Theorems to obtain geometric inequalities, focusing on the Extended Minkowski Inequality.

The discussion in this paper focuses on how boundary based smooth shape design can be carried out. For this we treat surface generation as a mathematical boundary-value problem. In particular, we utilize elliptic Partial Differential Equations (PDEs) of arbitrary order. Using the methodology outlined here a designer can therefore generate the geometry of shapes satisfying an arbitrary set of boundary conditions. The boundary conditions for the chosen PDE can be specified as curves in 3-space defining the profile geometry of the shape. We show how a compact analytic solution for the chosen arbitrary order PDE can be formulated enabling complex shapes to be designed and manipulated in real time. This solution scheme, although analytic, satisfies exactly, even in the case of general boundary conditions, where the resulting surface has a closed form representation allowing real time shape manipulation. In order to enable users to appreciate the powerful shape design and manipulation capability of the method, we present a set of practical examples