A thesis submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Doctor of Philosophy, Johannesburg, 2013. / The (1+3) dimensional wave and Klein-Gordon equations are constructed using the covariant d'Alembertian operator on several spacetimes of interest. Equations on curved geometry inherit the nonlinearities of the geometry. These equations display interesting properties in a number of ways. In particular, the number of symmetries and therefore, the conservation laws reduce depending on how curved the manifold is. We study the symmetry properties and conservation laws of wave equations on Freidmann-Robertson-Walker, Milne, Bianchi, and de Sitter universes. Symmetry structures are used to reduce the number of unknown functions, and hence contribute to nding exact solutions of the equations. As expected, properties of reduction procedures using symmetries, variational structures and conservation laws are more involved than on the well known at (Minkowski) manifold.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy / In this thesis we used the following four types of optimal control problems: (i) Problems governed by systems of ordinary differential equations; (ii) Problems governed by systems of ordinary differential equations with time-delayed arguments appearing in both the state and the control variables; (iii) Problems governed by linear systems subject to sudden jumps in parameter values; (iv) A chemical reactor problem governed by a couple of nonlinear diffusion equations. • The aim of this thesis is to devise computational algorithms for solving the optimal control problems under consideration. However, our main emphasis are on the mathematical theory underlying the techniques, the convergence properties of the algorithms and the efficiency of the algorithms. Chapters II and III deal with problems of the first type, Chapters IV and V deal with problems of the second type, and Chapters VI and VII deal with problems of the third and fourth type respectively. A few numerical problems have been included in each of these Chapters to demonstrate the efficiency of the algorithms involved. The class of optimal control problems considered in Chapter II consists of a nonlinear system, a nonlinear cost functional, initial equality constraints, and terminal equality constraints. A Sequential Gradient-Restoration Algorithm is used to devise an iterative algorithm for solving this class of problems. 'I'he convergence properties of the algorithm are investigated. The class of optimal control problems considered in Chapter III consists of a nonlinear system, a nonlinear cost functional, and terminal as well as interior points equality constraints. The technique of control parameterization and Liapunov concepts are used to solve this class of problems, A computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints or inequality type was developed by Rei. 103 in 1989. In Chapter IV, we extend the results of Ref. 103 to a more general class of constrained time-delayed optimal control problems, which involves terminal state equality constraints, as well as terminal state inequality constraints and continuous state inequality constraints. In Ref. 104, a computational scheme using the technique of control parameterization was developed for solving a class of optimal control problems in which the cost functional includes the full variation of control. Chapter V is a straightforward extension of Ref. 104 to the time-delayed case. However the main contribution of this chapter is that many numerical examples have been solved. In Chapter VI, a class of linear systems subject to sudden jumps in parameter values is considered. To solve this class of stochastic control problem, we try to seek for the best feedback control law depending only on the measurable output. Based on this idea, we convert the original problem into an approximate constrained deterministic optimization problem, which can be easily solved by any existing nonlinear programming technique. In Chapter VII, a chemical reactor problem and its control to achieve a desired output temperature is considered. A finite element Galerkin method is used to convert the original distributed optimal control problem into a quadratic programming problem with linear constraints, which can he solved by any standard quadratic programming software . / Andrew Chakane 2018

The 24 items of the original English version of the Social Justice Scale (Torres-Harding et al., 2012) were translated into Korean by four translators, who discussed and agreed upon consensus versions. Four different translators then back translated this version into English. The resulting Korean version of SJS (K-SJS) was completed by 537 adult native Korean speakers. Confirmatory factor analysis, exploratory factor analysis, and multi-group confirmatory factor analysis indicated that the K-SJS has high internal consistency, factors appropriately, fits the original model well, and demonstrates invariance across Korean men and women. Structural equation modeling indicated that the effects of attitude, perceived behavioral control, and subjective norms on behavioral intentions were positive and significant. In short, the K-SJS showed acceptable reliability and validity based on a large sample of South Korean adults and shows promise as a new tool to study social justice attitudes among Korean speakers.

Este trabalho contém um estudo sobre equações de ondas fracamente dissipativas definidas em domínios de fronteira móvel ∂2u/∂t2/ + η∂u/∂t - Δu + g(u) = f(x,t), (x,t) ∈ ^Dτ, onde ^Dτ = ∪t∈(τ,+ ∞) Ot X . Dizemos que domínio Dτ possui fronteira móvel se admitirmos que a fronteira Γt de de Ot varia em relação a t. Nossa contribuição é dividida em três etapas. 1 - Provamos que o problema munido da condição de fronteira de Dirichlet é bem posto no sentido de Hadamard (existência global, unicidade e dependência contínua dos dados) para soluções fortes e fracas. Nessa etapa utilizamos um método clássico que transforma o domínio dependente de t em um domínio fixo. Como consequência observamos que o sistema é essencialmente não autônomo. 2 - Buscamos uma teoria de sistemas dinâmicos não autônomos para estudar o operador solução do problema como um processo U(t; τ) : Xτ → Xτ, t≥ τ, definido em espaços de fase Xt = H01(Ot) × L2(Ot) que são dependentes do tempo t. 3 - No contexto da dinâmica de longo prazo encontramos hipóteses para garantir que o sistema dinâmico associado ao problema de ondas em domínios de fronteira móvel possui um atrator pullback. Basicamente admitimos que o domínio é crescente e \"time-like\". Salientamos que o nosso trabalho é o primeiro que estuda tais equações de ondas sob o ponto de vista de sistemas dinâmicos não-autônomos. Para equações parabólicas, resultados no mesmo contexto foram obtidos anteriormente por Kloeden, Marín-Rubio e Real [JDE 244 (2008) 2062-2090] e Kloeden, Real e Sun [JDE 246 (2009) 4702-4730]. Entretanto o nosso problema á hiperbólico e nã possui a regularidade das equações parabólicas. / In this work we study a weakly dissipative wave equation defined in domains with moving boundary ∂2u/∂t2/ + η∂u/∂t - Δu + g(u) = f(x,t), (x,t) ∈ Dτ, where D&tau> = ∪t∈(τ,+ ∞) Ot X . We says that a domain D&tau has moving boundary if the boundary &Gama;t of Ot varies with respect to t. Our contribution is threefold. 1 - We prove that the wave equation equipped with Dirichlet boundary condition is well-posed in the sense of Hadamard (global existence, uniqueness and continuous dependence with respect to data) for weak and strong solutions. This is done by using a classical argument that transforms the time dependent domain in a fixed domain. As a consequence we see that the problem is essentially non-autonomous. 2 -We find a theory of non-autonomous dynamical systems in order to study the solution operator as a process U(t; τ) : Xτ → Xsub>t, t≥τ, defined in time dependent phase spaces Xt = H01 (Ot) × L2.(Ot. 3 - In the context of long-time behavior of solutions we find suitable conditions to guarantee the existence of a pullback attractor. Roughly speaking, we assume the domain Q is expanding and time-like. We emphasize that our work is the first one that consider wave equations in noncylindrical domains as non-autonomous dynamical systems. With respect to parabolic equations, similar results were early obtained by Kloeden, Marín-Rubio and Real [JDE 244 (2008) 2062-2090] and Kloeden, Real and Sun [JDE 246 (2009) 4702-4730]. However our problem is hyperbolic and does not enjoy regularity properties as the parabolic ones.

by Tong Shiu Sing Dominic. / Parallel title in Chinese characters. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 236-241). / Acknowledgements --- p.iv / Abstract --- p.v / Chapter 1 --- Open Systems and Quasinormal Modes --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.1.1 --- Non-Hermitian Systems --- p.1 / Chapter 1.1.2 --- Optical Cavities as Open Systems --- p.3 / Chapter 1.1.3 --- Outline of this Thesis --- p.6 / Chapter 1.2 --- Simple Models of Open Systems --- p.10 / Chapter 1.3 --- Contributions of the Author --- p.14 / Chapter 2 --- Completeness and Orthogonality --- p.16 / Chapter 2.1 --- Introduction --- p.16 / Chapter 2.2 --- Green's Function of the Open System --- p.19 / Chapter 2.3 --- High Frequency Behaviour of the Green's Function --- p.24 / Chapter 2.4 --- Completeness of Quasinormal Modes --- p.29 / Chapter 2. 5 --- Method of Projection --- p.31 / Chapter 2.5.1 --- Problems with the Usual Method of Projection --- p.31 / Chapter 2.5.2 --- Modified Method of Projection --- p.33 / Chapter 2.6 --- Uniqueness of Representation --- p.38 / Chapter 2.7 --- Definition of Inner Product and Quasi-Stationary States --- p.39 / Chapter 2.7.1 --- Orthogonal Relation of Quasinormal Modes --- p.39 / Chapter 2.7.2 --- Definition of Hilbert Space and State Vectors --- p.41 / Chapter 2.8 --- Hermitian Limits --- p.43 / Chapter 2.9 --- Numerical Examples --- p.45 / Chapter 3 --- Time-Independent Perturbation --- p.58 / Chapter 3.1 --- Introduction --- p.58 / Chapter 3.2 --- Formalism --- p.60 / Chapter 3.2.1 --- Expansion of the Perturbed Quasi-Stationary States --- p.60 / Chapter 3.2.2 --- Formal Solution --- p.62 / Chapter 3.2.3 --- Perturbative Series --- p.66 / Chapter 3.3 --- Diagrammatic Perturbation --- p.70 / Chapter 3.3.1 --- Series Representation of the Green's Function --- p.70 / Chapter 3.3.2 --- Eigenfrequencies --- p.73 / Chapter 3.3.3 --- Eigenfunctions --- p.75 / Chapter 3.4 --- Numerical Examples --- p.77 / Chapter 4 --- Method of Diagonization --- p.81 / Chapter 4.1 --- Introduction --- p.81 / Chapter 4.2 --- Formalism --- p.82 / Chapter 4.2.1 --- Matrix Equation with Non-unique Solution --- p.82 / Chapter 4.2.2 --- Matrix Equation with a Unique Solution --- p.88 / Chapter 4.3 --- Numerical Examples --- p.91 / Chapter 5 --- Evolution of the Open System --- p.97 / Chapter 5.1 --- Introduction --- p.97 / Chapter 5.2 --- Evolution with Arbitrary Initial Conditions --- p.99 / Chapter 5.3 --- Evolution with the Outgoing Plane Wave Condition --- p.106 / Chapter 5.3.1 --- Evolution Inside the Cavity --- p.106 / Chapter 5.3.2 --- Evolution Outside the Cavity --- p.110 / Chapter 5.4 --- Physical Implications --- p.112 / Chapter 6 --- Time-Dependent Perturbation --- p.114 / Chapter 6.1 --- Introduction --- p.114 / Chapter 6.2 --- Inhomogeneous Wave Equation --- p.117 / Chapter 6.3 --- Perturbative Scheme --- p.120 / Chapter 6.4 --- Energy Changes due to the Perturbation --- p.128 / Chapter 6.5 --- Numerical Examples --- p.131 / Chapter 7 --- Adiabatic Approximation --- p.150 / Chapter 7.1 --- Introduction --- p.150 / Chapter 7.2 --- The Effect of a Varying Refractive Index --- p.153 / Chapter 7.3 --- Adiabatic Expansion --- p.156 / Chapter 7.4 --- Numerical Examples --- p.167 / Chapter 8 --- Generalization of the Formalism --- p.176 / Chapter 8. 1 --- Introduction --- p.176 / Chapter 8.2 --- Generalization of the Orthogonal Relation --- p.180 / Chapter 8.3 --- Evolution with the Outgong Wave Condition --- p.183 / Chapter 8.4 --- Uniform Convergence of the Series Representation --- p.193 / Chapter 8.5 --- Uniqueness of Representation --- p.200 / Chapter 8.6 --- Generalization of Standard Calculations --- p.202 / Chapter 8.6.1 --- Time-Independent Perturbation --- p.203 / Chapter 8.6.2 --- Method of Diagonization --- p.206 / Chapter 8.6.3 --- Remarks on Dynamical Calculations --- p.208 / Appendix A --- p.209 / Appendix B --- p.213 / Appendix C --- p.225 / Appendix D --- p.231 / Appendix E --- p.234 / References --- p.236

Lee Joy Yan Agatha. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [112]-114). / Text in English; abstracts in English and Chinese. / Lee Joy Yan Agatha. / Abstract --- p.ii / Acknowledgement --- p.iii / Contents --- p.iv / List of Figures --- p.viii / List of Tables --- p.xii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- The Transfer Integral Equation Method --- p.3 / Chapter 2.1 --- The System --- p.3 / Chapter 2.1.1 --- The Hamiltonian --- p.4 / Chapter 2.1.2 --- The length parameter --- p.5 / Chapter 2.1.3 --- The temperature parameter --- p.5 / Chapter 2.2 --- The Transfer Integral Equation --- p.6 / Chapter 2.2.1 --- The partition function --- p.6 / Chapter 2.2.2 --- The transfer integral equation --- p.6 / Chapter 2.2.3 --- The pseudo-Schrodinger equation approximation --- p.7 / Chapter 2.2.4 --- Distribution function of the displacements --- p.9 / Chapter 2.3 --- The Thermodynamics --- p.10 / Chapter 2.3.1 --- Internal energy and heat capacity --- p.10 / Chapter 2.3.2 --- Displacement fluctuation --- p.12 / Chapter 2.3.3 --- Displacement correlation function --- p.12 / Chapter Chapter 3. --- The Φ4 Chain --- p.14 / Chapter 3.1 --- Soliton In The Chain --- p.15 / Chapter 3.1.1 --- Kink soliton and antikink soliton --- p.15 / Chapter 3.1.2 --- Energy of a static kink --- p.18 / Chapter 3.2 --- Low Temperature WKB Approximation for the Φ4 Chain --- p.20 / Chapter 3.2.1 --- The ground state energy ε0 and tunneling-splitting contribution --- p.20 / Chapter 3.2.2 --- First order WKB approximation of ΨRo( φ) --- p.22 / Chapter 3.2.3 --- Second order WKB wavefunction ΨRo( φ)） --- p.26 / Chapter 3.2.4 --- Third order WKB wavefunction for ΨRo( φ) --- p.27 / Chapter 3.3 --- Thermodynamics --- p.28 / Chapter 3.3.1 --- Ground state energy ε0 and wavefunction Ψo( φ) --- p.28 / Chapter 3.3.2 --- Internal energy and heat capacity --- p.33 / Chapter 3.3.3 --- Displacement correlation function --- p.36 / Chapter Chapter 4. --- Other Nonlinear Klein-Gordon Models --- p.42 / Chapter 4.1 --- The φ8 Chain --- p.42 / Chapter 4.1.1 --- The potential --- p.42 / Chapter 4.1.2 --- The ground state energy εo and wavefunction Ψo( φ) --- p.44 / Chapter 4.1.3 --- Internal energy and heat capacity --- p.49 / Chapter 4.1.4 --- Displacement correlation function cyy(n) --- p.51 / Chapter 4.2 --- The Gaussian-Double-Well Chains --- p.53 / Chapter 4.2.1 --- The potential --- p.53 / Chapter 4.2.2 --- The ground state energy εo and wavefunction ψo --- p.55 / Chapter 4.2.3 --- Internal energy and heat capacity --- p.58 / Chapter 4.2.4 --- Displacement correlation function cyy(n) --- p.59 / Chapter 4.3 --- Comparison Between Different NKG Models --- p.61 / Chapter 4.3.1 --- The potentials --- p.61 / Chapter 4.3.2 --- Ground state energy εo and wavefunction ψo(ψ) --- p.65 / Chapter 4.3.3 --- Internal energy and heat capacity --- p.68 / Chapter 4.3.4 --- Displacement fluctuation --- p.70 / Chapter 4.3.5 --- Displacement correlation function cyy(n) --- p.71 / Chapter 4.4 --- Linear Response of a NKG Chain to a Static Perturbing Field --- p.75 / Chapter 4.4.1 --- The external perturbing field --- p.75 / Chapter 4.4.2 --- The linear response --- p.75 / Chapter 4.4.3 --- Linear response of an array of weakly coupled NKG chains --- p.80 / Chapter Chapter 5. --- Quantum Corrections --- p.86 / Chapter 5.1 --- The Effective Potential --- p.86 / Chapter 5.1.1 --- The smearing parameter --- p.86 / Chapter 5.1.2 --- The effective potential --- p.88 / Chapter 5.2 --- Quantum Corrections on Thermodynamics --- p.90 / Chapter 5.2.1 --- The ground state energy εo and wavefunction ψo(ψ) --- p.90 / Chapter 5.2.2 --- The heat capacity --- p.94 / Chapter 5.2.3 --- Displacement correlation function and displacement fluctuation --- p.97 / Chapter Chapter 6. --- Conclusion --- p.103 / Appendix A. Infinite-Square-Well Basis Diagonalization --- p.105 / Appendix B. Oscillator Basis Diagonalization --- p.110 / Bibliography --- p.112

by Ho Yum Bun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 93-94). / Text in English; abstracts in English and Chinese. / by Ho Yum Bun. / List of Figures --- p.3 / Abstract --- p.9 / Chinese Abstract --- p.10 / Acknowledgement --- p.11 / Chapter 1 --- Introduction --- p.12 / Chapter 1.1 --- Motivation --- p.12 / Chapter 1.2 --- What is Sonoluminescence? --- p.13 / Chapter 1.3 --- The Main Task of this Project --- p.13 / Chapter 1.4 --- Organization of this Thesis --- p.13 / Chapter 2 --- Reviews on One-dimensional Dynamical Cavity Problem --- p.15 / Chapter 2.1 --- Introduction --- p.15 / Chapter 2.2 --- Formulation --- p.15 / Chapter 2.3 --- Moore's R Function Method --- p.18 / Chapter 2.4 --- Mode Expansion Method --- p.19 / Chapter 2.5 --- Transformation method --- p.20 / Chapter 2-6 --- Summary --- p.21 / Chapter 3 --- Numerical Results For One-dimensional Dynamical Cavity Prob- lem --- p.22 / Chapter 3.1 --- Introduction --- p.22 / Chapter 3.2 --- Evolution of a Cavity System --- p.23 / Chapter 3.3 --- Motion of the Moving Mirror --- p.23 / Chapter 3.4 --- R(z) Function --- p.24 / Chapter 3.4.1 --- Construction of R(z) Function --- p.24 / Chapter 3.4.2 --- Numerical R(z) Function --- p.27 / Chapter 3.5 --- Results --- p.27 / Chapter 3.5.1 --- Results with Moore's R(z) Function Method --- p.27 / Chapter 3.5.2 --- Results with the Mode Expansion Method --- p.29 / Chapter 3.5.3 --- Results with the Transformation Method --- p.36 / Chapter 3.6 --- Summary --- p.36 / Chapter 4 --- Spherical Dynamical Cavity Problem --- p.37 / Chapter 4.1 --- Introduction --- p.37 / Chapter 4.2 --- Formulation --- p.37 / Chapter 4.3 --- Motion of a Moving Spherical Mirror --- p.39 / Chapter 4.4 --- Summary --- p.40 / Chapter 5 --- The G(z) Function Method --- p.41 / Chapter 5.1 --- Introduction --- p.41 / Chapter 5.2 --- G(z) Function --- p.42 / Chapter 5.2.1 --- Ideas of Deriving the G(z) Function --- p.42 / Chapter 5.2.2 --- Formalism --- p.42 / Chapter 5.2.3 --- Initial G(z) Function --- p.45 / Chapter 5.3 --- Construction of the G(z) Function --- p.46 / Chapter 5.3.1 --- Case I : l=0 --- p.46 / Chapter 5.3.2 --- Case II : l > 0 --- p.49 / Chapter 5.4 --- Asymptotic Series Solution of G(z) --- p.50 / Chapter 5.5 --- Application to Resonant Mirror Motion --- p.52 / Chapter 5.6 --- Regularization of G(z) --- p.58 / Chapter 5.7 --- Behaviors of the Fields --- p.58 / Chapter 5.7.1 --- z vs tf Graph --- p.61 / Chapter 5.7.2 --- Case 1: l= 0 --- p.61 / Chapter 5.7.3 --- "Case2: l= 1,2" --- p.62 / Chapter 5.7.4 --- Case 3: l= 3 --- p.73 / Chapter 5.7.5 --- Section Summary --- p.73 / Chapter 5.8 --- Summary --- p.73 / Chapter 6 --- Three-dimensional Mode Expansion Method and Transforma- tion Method --- p.75 / Chapter 6.1 --- Introduction --- p.75 / Chapter 6.2 --- Mode Expansion Method --- p.75 / Chapter 6.2.1 --- Formalism --- p.75 / Chapter 6.2.2 --- Application of Floquet's Theory --- p.78 / Chapter 6.2.3 --- Results --- p.80 / Chapter 6.3 --- The Transformation Method --- p.80 / Chapter 6.3.1 --- The Method --- p.80 / Chapter 6.3.2 --- Numerical Schemes --- p.86 / Chapter 6.3.3 --- Results --- p.89 / Chapter 6.4 --- Summary --- p.89 / Chapter 7 --- Conclusion --- p.90 / Chapter 7.1 --- The One-dimensional Dynamical Cavity Problem --- p.90 / Chapter 7.2 --- The Dynamical Spherical Cavity Problem --- p.91 / Chapter 7.3 --- Numerical Methods --- p.91 / Chapter 7.4 --- Further Investigation --- p.92 / Bibliography --- p.93

by Chow Ha Tak. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 96-97). / Abstract also in Chinese. / Chapter 1 --- The Laplacian on a Riemannian Manifold --- p.5 / Chapter 1.1 --- Riemannian metrics --- p.5 / Chapter 1.2 --- L2 Spaces of Functions and Forms --- p.6 / Chapter 1.3 --- The Laplacian on Functions and Forms --- p.8 / Chapter 2 --- Hodge Theory for Functions and Forms --- p.14 / Chapter 2.1 --- Analytic Preliminaries --- p.14 / Chapter 2.2 --- The Hodge Theorem for Functions --- p.20 / Chapter 2.3 --- The Hodge Theorem for Forms --- p.27 / Chapter 2.4 --- Regularity Results --- p.29 / Chapter 2.5 --- The Kernel of the Laplacian on Forms --- p.33 / Chapter 3 --- Fermion Calculus and Weitzenbock Formula --- p.36 / Chapter 3.1 --- The Levi-Civita Connection --- p.36 / Chapter 3.2 --- Fermion calculus --- p.39 / Chapter 3.3 --- "Weitzenbock Formula, Bochner Formula and Garding's Inequality" --- p.53 / Chapter 3.4 --- The Laplacian in Exponential Coordinates --- p.59 / Chapter 4 --- The Construction of the Heat Kernel --- p.63 / Chapter 4.1 --- Preliminary Results for the Heat Kernel --- p.63 / Chapter 4.2 --- Construction of the Heat Kernel --- p.66 / Chapter 4.2.1 --- Construction of the Parametrix --- p.66 / Chapter 4.2.2 --- The Heat Kernel for Functions --- p.70 / Chapter 4.2.3 --- The Heat Kernel for Forms --- p.76 / Chapter 4.3 --- The Asymptotics of the Heat Kernel --- p.77 / Chapter 5 --- The Heat Equation Approach to the Chern-Gauss- Bonnet Theorem --- p.82 / Chapter 5.1 --- The Heat Equation Approach --- p.82 / Chapter 5.2 --- Proof of the Chern-Gauss-Bonnet Theorem --- p.85 / Chapter 5.3 --- Introduction to Atiyah-Singer Index Theorem --- p.87 / Chapter 5.3.1 --- A Survey of Characteristic Forms --- p.87 / Chapter 5.3.2 --- The Hirzenbruch Signature Theorem --- p.90 / Chapter 5.3.3 --- The Atiyah-Singer Index Theorem --- p.93 / Bibliography / Notation index

This thesis considers one-dimensional moving mesh (MM) methods coupled with semi-Lagrangian (SL) discretisations of partial differential equations (PDEs) for meteorological applications. We analyse a semi-Lagrangian numerical solution to the viscous Burgers’ equation when using linear interpolation. This gives expressions for the phase and shape errors of travelling wave solutions which decay slowly with increasing spatial and temporal resolution. These results are verified numerically and demonstrate qualitative agreement for high order interpolants. The semi-Lagrangian discretisation is coupled with a 1D moving mesh, resulting in a moving mesh semi-Lagrangian (MMSL) method. This is compared against two moving mesh Eulerian methods, a two-step remeshing approach, solved with the theta-method, and a coupled moving mesh PDE approach, which is solved using the MATLAB solver ODE45. At each time step of the SL method, the mesh is updated using a curvature based monitor function in order to reduce the interpolation error, and hence numerical viscosity. This MMSL method exhibits good stability properties, and captures the shape and speed of the travelling wave well. A meteorologically based 1D vertical column model is described with its SL solution procedure. Some potential benefits of adaptivity are demonstrated, with static meshes adapted to initial conditions. A moisture species is introduced into the model, although the effects are limited.

The growing fields of spintronics and nanotechnology have created increased interest in developing the means to manipulate the spin of electrons. One such method arises from the combination of the spin-orbit interaction and the broken inversion symmetry that arises at surfaces and interfaces, and has prompted many recent investigations on metallic surfaces. A method by which surface states, in the absence of spin orbit effects, have been successfully investigated is the Green function embedding scheme of Inglesfield. This has been integrated into a self consistent FLAPW density functional framework based on the scalar relativistic K¨olling Harmon equation. Since the spin of the electron is a direct effect of special relativity, calculations involving the spin orbit interaction are best performed using solutions of the Dirac equation. This work describes the extension of Green’s function embedding to include the Dirac equation and how fully relativistic FLAPW surface electronic structure calculations are implemented. The general procedure used in performing a surface calculation in the scalar relativistic case is closely followed. A bulk transfer matrix is defined and used to generate the complex band structure and an embedding potential. This embedding potential is then used to produce a self consistent surface potential, leading to a Green’s function from which surface state dispersions and splittings are calculated. The bulk embedding potential can also be employed in defining channel functions and these provide a natural framework in which to explore transport properties. A relativistic version of a well known expression for the ballistic conductance across a device is derived in this context. Differences between the relativistic and nonrelativistic methods are discussed in detail. To test the validity of the scheme, a fully relativistic calculation of the extensively studied spin orbit split L-gap surface state on Au(111) is performed, which agrees well with experiment and previous calculations. Contributions to the splitting from different angular momentum channels are also provided. The main advantages of the relativistic embedding method are the full inclusion of the spin orbit interaction to all orders, the true semi infinite nature of the technique, allowing the full complex bands of the bulk crystal to be represented and the fact that a only small number of surface layers is needed in comparison to other existing methods.