This paper is concerned with finite difference methods for approximating solutions to the partial differential heat equation. The first chapel gives some introductory background into the physical problem, then motivates three finite difference methods. Chapters II through IV provide statements and proofs for the theorems used in the methods of Chapter I. The final Chapter, V, provides conclusions and an indication of future work. An appendix includes the computer codes written by the author with numerical results.

Wave-based models of sound propagation can be used to predict and synthesize sounds as they would be heard naturally in room acoustic environments. The numerical simulation of such models with traditional time-stepping grid-based methods can be an expensive process, due to the sheer size of listening environments (e.g., auditoriums and concert halls) and due to the temporal resolution required by audio rates that resolve frequencies up to the limit of human hearing. Finite difference methods comprise a simple starting point for such simulations, but they are known to suffer from approximation errors that may necessitate expensive grid refinements in order to achieve sufficient levels of accuracy. As such, a significant amount of research has gone into designing finite difference methods that are highly accurate while remaining computationally efficient. The problem of designing and using accurate finite difference schemes is compounded by the fact that room acoustics models require complex boundary conditions to model frequency-dependent wall impedances over non-trivial geometries. The implementation of such boundary conditions in a numerically stable manner has been a challenge for some time. Stable boundary conditions for finite difference room acoustics simulations have been formulated in the past, but generally they have only been useful in modelling trivial geometries (e.g., idealised shoebox halls). Finite volume methods have recently been shown to be a viable solution to the problem of complex boundary conditions over non-trivial geometries, and they also allow for the use of energy methods for numerical stability analyses. Finite volume methods lend themselves naturally to fully unstructured grids and they can simplify to the types of grids typically used in finite difference methods. This allows for room acoustics simulation models that balance the simplicity of finite difference methods for wave propagation in air with the detail of finite volume methods for the modelling of complex boundaries. This thesis is an exploration of these two distinct, yet related, approaches to wave-based room acoustic simulations. The overarching theme in this investigation is the balance between accuracy, computational efficiency, and numerical stability. Higher-order and optimised schemes in two and three spatial dimensions are derived and compared, towards the goal of finding accurate and efficient finite difference schemes. Numerical stability is analysed using frequency-domain analyses, as well as energy techniques whenever possible, allowing for stable and frequency-dependent boundary conditions appropriate for room acoustics modelling. Along the way, the use of non-Cartesian grids is investigated, geometric relationships between certain finite difference and finite volume schemes are explored, and some problems associated to staircasing effects at boundaries are considered. Also, models of sound absorption in air are incorporated into these numerical schemes, using physical parameters that are appropriate for room acoustic scenarios.

This thesis studies the physics of multilayer heterostructures grown from topological insulators (TIs), primarily bismuth selenide and antimony telluride, and other topologically trivial materials. This is done by extending a standard low-energy 3D TI Hamiltonian and varying its associated material parameters across the simulation domain. New results arising from the position-dependent TI interface model are found. For the first time, this method is incorporated into a density-functional theory (DFT) solver in order to study the self-consistent charge density in multilayer TI heterostructures due to the interface states. The thesis is structured as follows. The introduction (Ch. 1) presents a pedagogical review of the theory of 3D TIs and low-energy Hamiltonians used to study them, as well as typical methods in solid state physics that are made use of throughout the thesis. Chapter 2 presents the position-dependent Hamiltonian, showing new evidence for topological features of bulk states including varying degrees of band mixing and inversion; also, interface state tunnelling is shown to be affected by atomic layer orbital overlap, and incomplete localisation of surface states is demonstrated for antimony telluride. Chapter 3 presents a new DFT model of TI heterostructure interfaces and shows how conduction through TI interface states can be controlled with an electric field. Chapter 4 covers the extension of the model in Ch. 1 to 2D cross-sections of TI wires and heterostructures, showing for the first time evidence of localisation of conduction almost entirely within the inner interfaces of a 2D heterostructure wire. Chapter 5 presents our work with magnetic fields, demonstrating evolution of interface and bulk states with changing magnetic field and Landau level, as well as presenting new evidence for more complex spin structures in bismuth selenide arising from Landé factor signs. Our conclusions are presented in Chapter 6.

Partial differential equations (PDEs) are used to model various phenomena in nature and society, ranging from the motion of fluids and electromagnetic waves to the stock market and traffic jams. There are many methods for numerically approximating solutions to PDEs. Some of the most commonly used ones are the finite volume method, the finite element method, and the finite difference method. All methods have their strengths and weaknesses, and it is the problem at hand that determines which method that is suitable. In this thesis, we focus on the finite difference method which is conceptually easy to understand, has high-order accuracy, and can be efficiently implemented in computer software. We use the finite difference method on summation-by-parts (SBP) form, together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT). Together, SBP and SAT provide a technique for overcoming most of the drawbacks of the finite difference method. The SBP-SAT technique can be used to derive energy stable schemes for any linearly well-posed initial boundary value problem. The stability is not restricted by the order of accuracy, as long as the numerical scheme can be written in SBP form. The weak boundary conditions can be extended to interfaces which are used either in domain decomposition for geometric flexibility, or for coupling of different physics models. The contributions in this thesis are twofold. The first part, papers I-IV, develops stable boundary and interface procedures for computational fluid dynamics problems, in particular for problems related to the Navier-Stokes equations and conjugate heat transfer. The second part, papers V-VI, utilizes duality to construct numerical schemes which are not only energy stable, but also dual consistent. Dual consistency alone ensures superconvergence of linear integral functionals from the solutions of SBP-SAT discretizations. By simultaneously considering well-posedness of the primal and dual problems, new advanced boundary conditions can be derived. The new duality based boundary conditions are imposed by SATs, which by construction of the continuous boundary conditions ensure energy stability, dual consistency, and functional superconvergence of the SBP-SAT schemes.

In this thesis, high order accurate discretization schemes for partial differential equations are investigated. In the first paper, the linearized two-dimensional Navier-Stokes equations are considered. A special formulation of the boundary conditions is used and estimates for the solution to the continuous problem in terms of the boundary conditions are derived using a normal mode analysis. Similar estimates are achieved for the discretized equations. For the discretization, a second order finite difference scheme on a staggered mesh is used. In Paper II, the analysis for the second order scheme is used to develop a fourth order scheme for the fully nonlinear Navier-Stokes equations. The fully nonlinear incompressible Navier-Stokes equations in two space dimensions are considered on an orthogonal curvilinear grid. Numerical tests are performed with a fourth order accurate Padé type spatial finite difference scheme and a semi-implicit BDF2 scheme in time. In Papers III-V, a class of high order accurate time-discretization schemes based on the deferred correction principle is investigated. The deferred correction principle is based on iteratively eliminating lower order terms in the local truncation error, using previously calculated solutions, in each iteration obtaining more accurate solutions. It is proven that the schemes are unconditionally stable and stability estimates are given using the energy method. Error estimates and smoothness requirements are derived. Special attention is given to the implementation of the boundary conditions for PDE. The scheme is applied to a series of numerical problems, confirming the theoretical results. In the sixth paper, a time-compact fourth order accurate time discretization for the one- and two-dimensional wave equation is considered. Unconditional stability is established and fourth order accuracy is numerically verified. The scheme is applied to a two-dimensional wave propagation problem with discontinuous coefficients.

In this thesis, the numerical solution of time-dependent partial differential equations (PDE) is studied. In particular high-order finite difference methods on Summation-by-parts (SBP) form are analysed and applied to model problems as well as the PDEs governing aerodynamics. The SBP property together with an implementation of boundary conditions called SAT (Simultaneous Approximation Term), yields stability by energy estimates. The first derivative SBP operators were originally derived for Cartesian grids. Since aerodynamic computations are the ultimate goal, the scheme must also be stable on curvilinear grids. We prove that stability on curvilinear grids is only achieved for a subclass of the SBP operators. Furthermore, aerodynamics often requires addition of artificial dissipation and we derive an SBP version. With the SBP-SAT technique it is possible to split the computational domain into a multi-block structure which simplifies grid generation and more complex geometries can be resolved. To resolve extremely complex geometries an unstructured discretisation method must be used. Hence, we have studied a finite volume approximation of the Laplacian. It can be shown to be on SBP form and a new boundary treatment is derived. Based on the Laplacian scheme, we also derive an SBP artificial dissipation for finite volume schemes. We derive a new set of boundary conditions that leads to an energy estimate for the linearised three-dimensional Navier-Stokes equations. The new boundary conditions will be used to construct a stable SBP-SAT discretisation. To obtain an energy estimate for the discrete equation, it is necessary to discretise all the second derivatives by using the first derivative approximation twice. According to previous theory that would imply a degradation of formal accuracy but we present a proof that this is not the case.

In this thesis, stable and efficient hybrid methods which combine high order finite difference methods and unstructured finite volume methods for time-dependent initial boundary value problems have been developed. The hybrid methods make it possible to combine the efficiency of the finite difference method and the flexibility of the finite volume method. We carry out a detailed analysis of the stability of the hybrid methods, and in particular the stability of interface treatments between structured and unstructured blocks. Both the methods employ so called summation-by-parts operators and impose boundary and interface conditions weakly, which lead to an energy estimate and stability. We have constructed and analyzed first-, second- and fourth-order Laplacian based artificial dissipation operators for finite volume methods on unstructured grids. The first-order artificial dissipation can handle shock waves, and the fourth-order artificial dissipation eliminates non-physical numerical oscillations efficiently. A stable hybrid method for hyperbolic problems has been developed. It is shown that the stability at the interface can be obtained by modifying the dual grid of the unstructured finite volume method close to the interface. The hybrid method is applied to the Euler equation by the coupling of two stand-alone CFD codes. Since the coupling is administered by a third separate coupling code, the hybrid method allows for individual development of the stand-alone codes. It is shown that the hybrid method is an accurate, efficient and practically useful computational tool that can handle complex geometries and wave propagation phenomena. Stable and accurate interface treatments for the linear advection–diffusion equation have been studied. Accurate high-order calculation are achieved in multiple blocks with interfaces. Three stable interface procedures — the Baumann–Oden method, the “borrowing” method and the local discontinuous Galerkin method, have been investigated. The analysis shows that only minor differences separate the different interface handling procedures. A conservative stable and efficient hybrid method for a parabolic model problem has been developed. The hybrid method has been applied to the full Navier–Stokes equations. The numerical experiments support the theoretical conclusions and show that the interface coupling is stable and converges at the correct order for the Navier–Stokes equations.

The investigation of the dynamics of chemical reactions, both from the theoretical and experimental side, has drawn an increasing interest since Ahmed H. Zewail was awarded the 1999 Nobel Prize in Chemistry for his work on femtochemistry. On the experimental side, new techniques such as femtosecond lasers and attosecond lasers enable laser control of chemical reactions. Numerical simulations serve as a valuable complement to experimental techniques, not only for validation of experimental results, but also for simulation of processes that cannot be investigated through experiments. With increasing computer capacity, more and more physical phenomena fall within the range of what is possible to simulate. Also, the development of new, efficient numerical methods further increases the possibilities. The focus of this thesis is twofold; numerical methods for chemical reactions including dissociative states and methods that can deal with coexistence of spatial regions with very different physical properties. Dissociative chemical reactions are reactions where molecules break up into smaller components. The dissociation can occur spontaneously, e.g. by radioactive decay, or be induced by adding energy to the system, e.g. in terms of a laser field. Quantities of interest can for instance be the reaction probabilities of possible chemical reactions. This thesis discusses a boundary treatment model based on the perfectly matched layer (PML) approach to accurately describe dynamics of chemical reactions including dissociative states. The limitations of the method are investigated and errors introduced by the PML are quantified. The ability of a numerical method to adapt to different scales is important in the study of more complex chemical systems. One application of interest is long-range molecules, where the atoms are affected by chemical attractive forces that lead to fast movement in the region where they are close to each other and exhibits a relative motion of the atoms that is very slow in the long-range region. A numerical method that allows for spatial adaptivity is presented, based on the summation-by-parts-simultaneous approximation term (SBP-SAT) methodology. The accuracy and the robustness of the numerical method are investigated. / eSSENCE

In numerical simulations, problems stemming from aerodynamics pose many challenges for the method used. Some of these are addressed in this thesis, such as the fluid interacting with objects, the presence of shocks, and various types of boundary conditions. Scenarios of the kind mentioned above are described mathematically by initial boundary value problems (IBVPs). We discretize the IBVPs using high order accurate finite difference schemes on summation by parts form (SBP), combined with weakly imposed boundary conditions, a technique called simultaneous approximation term (SAT). By using the energy method, stability can be shown. The weak implementation is compared to the more commonly used strong implementation, and it is shown that the weak technique enhances the rate of convergence to steady state for problems with solid wall boundary conditions. The analysis is carried out for a linear problem and supported numerically by simulations of the fully non-linear Navier–Stokes equations. Another aspect of the boundary treatment is observed for fluid structure interaction problems. When exposed to eigenfrequencies, the coupled system starts oscillating, a phenomenon called flutter. We show that the strong implementation sometimes cause instabilities that can be mistaken for flutter. Most numerical schemes dealing with flows including shocks are first order accurate to avoid spurious oscillations in the solution. By modifying the SBP-SAT technique, a conservative and energy stable scheme is derived where the order of accuracy can be lowered locally. The new scheme is coupled to a shock-capturing scheme and it retains the high accuracy in smooth regions. For problems with complicated geometry, one strategy is to couple the finite difference method to the finite volume method. We analyze the accuracy of the latter on unstructured grids. For grids of bad quality the truncation error can be of zeroth order, indicating that the method is inconsistent, but we show that some of the accuracy is recovered. We also consider artificial boundary closures on unbounded domains. Non-reflecting boundary conditions for an incompletely parabolic problem are derived, and it is shown that they yield well-posedness. The SBP-SAT methodology is employed, and we prove that the discretized problem is stable.

[EN] In the stock markets, the process of estimating a fair price for a stock, option or commodity is consider the corner stone for this trade. There are several attempts to obtain a suitable mathematical model in order to enhance the estimation process for evaluating the options for short or long periods. The Black-Scholes partial differential equation (PDE) and its analytical solution, 1973, are considered a breakthrough in the mathematical modeling for the stock markets. Because of the ideal assumptions of Black-Scholes several alternatives have been developed to adequate the models to the real markets. Two strategies have been done to capture these behaviors; the first modification is to add jumps into the asset following Lévy processes, leading to a partial integro-differential equation (PIDE); the second is to allow the volatility to evolve stochastically leading to a PDE with two spatial variables. Here in this work, we solve numerically PIDEs for a wide class of Lévy processes using finite difference schemes for European options and also, the associated linear complementarity problem (LCP) for American option. Moreover, the models for options under stochastic volatility incorporated with jump-diffusion are considered. Numerical analysis for the proposed schemes is studied since it is the efficient and practical way to guarantee the convergence and accuracy of numerical solutions. In fact, without numerical analysis, careless computations may waste good mathematical models. This thesis consists of four chapters; the first chapter is an introduction containing historically review for stochastic processes, Black-Scholes equation and preliminaries on numerical analysis. Chapter two is devoted to solve the PIDE for European option under CGMY process. The PIDE for this model is solved numerically using two distinct discretization approximations; the first approximation guarantees unconditionally consistency while the second approximation provides unconditional positivity and stability. In the first approximation, the differential part is approximated using the explicit scheme and the integral part is approximated using the trapezoidal rule. In the second approximation, the differential part is approximated using the Patankar-scheme and the integral part is approximated using the four-point open type formula. Chapter three provides a unified treatment for European and American options under a wide class of Lévy processes as CGMY, Meixner and Generalized Hyperbolic. First, the reaction and convection terms of the differential part of the PIDE are removed using appropriate mathematical transformation. The differential part for European case is explicitly discretized , while the integral part is approximated using Laguerre-Gauss quadrature formula. Numerical properties such as positivity, stability and consistency for this scheme are studied. For the American case, the differential part of the LCP is discretized using a three-time level approximation with the same integration technique. Next, the Projected successive over relaxation and multigrid techniques have been implemented to obtain the numerical solution. Several numerical examples are given including discussion of the errors and computational cost. Finally in Chapter four, the PIDE for European option under Bates model is considered. Bates model combines both stochastic volatility and jump diffusion approaches resulting in a PIDE with a mixed derivative term. Since the presence of cross derivative terms involves the existence of negative coefficient terms in the numerical scheme deteriorating the quality of the numerical solution, the mixed derivative is eliminated using suitable mathematical transformation. The new PIDE is solved numerically and the numerical analysis is provided. Moreover, the LCP for American option under Bates model is studied. / [ES] El proceso de estimación del precio de una acción, opción u otro derivado en los mercados de valores es objeto clave de estudio de las matemáticas financieras. Se pueden encontrar diversas técnicas para obtener un modelo matemático adecuado con el fin de mejorar el proceso de valoración de las opciones para periodos cortos o largos. Históricamente, la ecuación de Black-Scholes (1973) fue un gran avance en la elaboración de modelos matemáticos para los mercados de valores. Es un modelo práctico para estimar el valor razonable de una opción. Sobre unos supuestos determinados, F. Black y M. Scholes obtuvieron una ecuación diferencial parcial lineal y su solución analítica. Desde entonces se han desarrollado modelos más complejos para adecuarse a la realidad de los mercados. Un tipo son los modelos con volatilidad estocástica que vienen descritos por una ecuación en derivadas parciales con dos variables espaciales. Otro enfoque consiste en añadir saltos en el precio del subyacente por medio de modelos de Lévy lo que lleva a resolver una ecuación integro-diferencial parcial (EIDP). En esta memoria se aborda la resolución numérica de una amplia clase de modelos con procesos de Lévy. Se desarrollan esquemas en diferencias finitas para opciones europeas y también para opciones americanas con su problema de complementariedad lineal (PCL) asociado. Además se tratan modelos con volatilidad estocástica incorporando difusión con saltos. Se plantea el análisis numérico ya que es el camino eficiente y práctico para garantizar la convergencia y precisión de las soluciones numéricas. De hecho, la ausencia de análisis numérico debilita un buen modelo matemático. Esta memoria está organizada en cuatro capítulos. El primero es una introducción con un breve repaso de los procesos estocásticos, el modelo de Black-Scholes así como nociones preliminares de análisis numérico. En el segundo capítulo se trata la EIDP para las opciones europeas según el modelo CGMY. Se proponen dos esquemas en diferencias finitas; el primero garantiza consistencia incondicional de la solución mientras que el segundo proporciona estabilidad y positividad incondicionales. Con el primer enfoque, la parte diferencial se discretiza por medio de un esquema explícito y para la parte integral se usa la regla del trapecio. En la segunda aproximación, para la parte diferencial se usa un esquema tipo Patankar y la parte integral se aproxima por medio de la fórmula de tipo abierto con cuatro puntos. En el capítulo tercero se propone un tratamiento unificado para una amplia clase de modelos de opciones en procesos de Lévy como CGMY, Meixner e hiperbólico generalizado. Se eliminan los términos de reacción y convección por medio de un apropiado cambio de variables. Después la parte diferencial se aproxima por un esquema explícito mientras que para la parte integral se usa la fórmula de cuadratura de Laguerre-Gauss. Se analizan positividad, estabilidad y consistencia. Para las opciones americanas, la parte diferencial del LCP se discretiza con tres niveles temporales mediante cuadratura de Laguerre-Gauss para la integración numérica. Finalmente se implementan métodos iterativos de proyección y relajación sucesiva y la técnica de multimalla. Se muestran varios ejemplos incluyendo estudio de errores y coste computacional. El capítulo 4 está dedicado al modelo de Bates que combina los enfoques de volatilidad estocástica y de difusión con saltos derivando en una EIDP con un término con derivadas cruzadas. Ya que la discretización de una derivada cruzada comporta la existencia de coeficientes negativos en el esquema que deterioran la calidad de la solución numérica, se propone un cambio de variables que elimina dicha derivada cruzada. La EIDP transformada se resuelve numéricamente y se muestra el análisis numérico. Por otra parte se estudia el LCP para opciones americanas con el modelo de Bates. / [CAT] El procés d'estimació del preu d'una acció, opció o un altre derivat en els mercats de valors és objecte clau d'estudi de les matemàtiques financeres . Es poden trobar diverses tècniques per a obtindre un model matemàtic adequat a fi de millorar el procés de valoració de les opcions per a períodes curts o llargs. Històricament, l'equació Black-Scholes (1973) va ser un gran avanç en l'elaboració de models matemàtics per als mercats de valors. És un model matemàtic pràctic per a estimar un valor raonable per a una opció. Sobre uns suposats F. Black i M. Scholes van obtindre una equació diferencial parcial lineal amb solució analítica. Des de llavors s'han desenrotllat models més complexos per a adequar-se a la realitat dels mercats. Un tipus és els models amb volatilitat estocástica que ve descrits per una equació en derivades parcials amb dos variables espacials. Un altre enfocament consistix a afegir bots en el preu del subjacent per mitjà de models de Lévy el que porta a resoldre una equació integre-diferencial parcial (EIDP) . En esta memòria s'aborda la resolució numèrica d'una àmplia classe de models baix processos de Lévy. Es desenrotllen esquemes en diferències finites per a opcions europees i també per a opcions americanes amb el seu problema de complementarietat lineal (PCL) associat. A més es tracten models amb volatilitat estocástica incorporant difusió amb bots. Es planteja l'anàlisi numèrica ja que és el camí eficient i pràctic per a garantir la convergència i precisió de les solucions numèriques. De fet, l'absència d'anàlisi numèrica debilita un bon model matemàtic. Esta memòria està organitzada en quatre capítols. El primer és una introducció amb un breu repàs dels processos estocásticos, el model de Black-Scholes així com nocions preliminars d'anàlisi numèrica. En el segon capítol es tracta l'EIDP per a les opcions europees segons el model CGMY. Es proposen dos esquemes en diferències finites; el primer garantix consistència incondicional de la solució mentres que el segon proporciona estabilitat i positivitat incondicionals. Amb el primer enfocament, la part diferencial es discretiza per mitjà d'un esquema explícit i per a la part integral s'empra la regla del trapezi. En la segona aproximació, per a la part diferencial s'usa l'esquema tipus Patankar i la part integral s'aproxima per mitjà de la fórmula de tipus obert amb quatre punts. En el capítol tercer es proposa un tractament unificat per a una àmplia classe de models d'opcions en processos de Lévy com ara CGMY, Meixner i hiperbòlic generalitzat. S'eliminen els termes de reacció i convecció per mitjà d'un apropiat canvi de variables. Després la part diferencial s'aproxima per un esquema explícit mentres que per a la part integral s'usa la fórmula de quadratura de Laguerre-Gauss. S'analitzen positivitat, estabilitat i consistència. Per a les opcions americanes, la part diferencial del LCP es discretiza amb tres nivells temporals amb quadratura de Laguerre-Gauss per a la integració numèrica. Finalment s'implementen mètodes iteratius de projecció i relaxació successiva i la tècnica de multimalla. Es mostren diversos exemples incloent estudi d'errors i cost computacional. El capítol 4 està dedicat al model de Bates que combina els enfocaments de volatilitat estocástica i de difusió amb bots derivant en una EIDP amb un terme amb derivades croades. Ja que la discretización d'una derivada croada comporta l'existència de coeficients negatius en l'esquema que deterioren la qualitat de la solució numèrica, es proposa un canvi de variables que elimina dita derivada croada. La EIDP transformada es resol numèricament i es mostra l'anàlisi numèrica. D'altra banda s'estudia el LCP per a opcions americanes en el model de Bates. / El-Fakharany, MMR. (2015). Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/53917 / TESIS