The spline approach to the numerical solution of parabolic partial differential equations

This thesis is concerned with the Numerical Solution of Partial Differential Equations. Initially some definitions and mathematical background are given, accompanied by the basic theories of solving linear systems and other related topics. Also, an introduction to splines, particularly cubic splines and their identities are presented. The methods used to solve parabolic partial differential equations are surveyed and classified into explicit or implicit (direct and iterative) methods. We concentrate on the Alternating Direction Implicit (ADI), the Group Explicit (GE) and the Crank-Nicolson (C-N) methods. A new method, the Splines Group Explicit Iterative Method is derived, and a theoretical analysis is given. An optimum single parameter is found for a special case. Two criteria for the acceleration parameters are considered; they are the Peaceman-Rachford and the Wachspress criteria. The method is tested for different numbers of both parameters. The method is also tested using single parameters, i. e. when used as a direct method. The numerical results and the computational complexity analysis are compared with other methods, and are shown to be competitive. The method is shown to have good stability property and achieves high accuracy in the numerical results. Another direct explicit method is developed from cubic splines; the splines Group Explicit Method which includes a parameter that can be chosen to give optimum results. Some analysis and the computational complexity of the method is given, with some numerical results shown to confirm the efficiency and compatibility of the method. Extensions to two dimensional parabolic problems are given in a further chapter. In this thesis the Dirichlet, the Neumann and the periodic boundary conditions for linear parabolic partial differential equations are considered. The thesis concludes with some conclusions and suggestions for further work

High-Order Numerical Methods in Lake Modelling

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint

The Du Fort and Frankel finite difference scheme applied to and adapted for a class of finance problems

We consider the finite difference method applied to a class of financial problems. Specifically, we investigate the properties of the Du Fort and Frankel finite difference scheme and experiment with adaptations of the scheme to improve on its consistency properties. The Du Fort and Frankel finite difference scheme is applied to a number of problems that frequently occur in finance. We specifically investigate problems associated with jumps, discontinuous behavior, free boundary problems and multi dimensionality. In each case we consider adaptations to the Du Fort and Frankel scheme in order to produce reliable results. CopyrightDissertation (MSc)--University of Pretoria, 2009.Mathematics and Applied Mathematicsunrestricte

Adaptive Space-Time Finite Element Methods for Optimization Problems Governed by Nonlinear Parabolic Systems

Subject of this work is the development of concepts for the efficient numerical solution of optimization problems governed by parabolic partial differential equations. Optimization problems of this type arise for instance from the optimal control of physical processes and from the identification of unknown parameters in mathematical models describing such processes. For their numerical treatment, these generically infinite-dimensional optimal control and parameter estimation problems have to be discretized by finite-dimensional approximations. This discretization process causes errors which have to be taken into account to obtain reliable numerical results. Focal point of the thesis at hand is the assessment of these discretization errors by a priori and especially a posteriori error analyses. Thereby, we consider Galerkin finite element discretizations of the state and the control variable in space and time. For the a priori analysis, we concentrate on the case of linear-quadratic optimal control problems. In this configuration, we prove error estimates of optimal order with respect to all involved discretization parameters. The a posteriori error estimation techniques are developed for a general class of nonlinear optimization problems. They provide separated and evaluable estimates for the errors caused by the different parts of the discretization and yield refinement indicators, which can be used for the automatic choice of suitable discrete spaces. The usage of adaptive refinement techniques within a strategy for balancing the several error contributions leads to efficient discretizations for the continuous problems. The presented results and developed concepts are substantiated by various numerical examples including large scale optimization problems motivated by concrete applications from engineering and chemistry

The Third Air Force/NASA Symposium on Recent Advances in Multidisciplinary Analysis and Optimization

The third Air Force/NASA Symposium on Recent Advances in Multidisciplinary Analysis and Optimization was held on 24-26 Sept. 1990. Sessions were on the following topics: dynamics and controls; multilevel optimization; sensitivity analysis; aerodynamic design software systems; optimization theory; analysis and design; shape optimization; vehicle components; structural optimization; aeroelasticity; artificial intelligence; multidisciplinary optimization; and composites

Kinematics and Robot Design II (KaRD2019) and III (KaRD2020)

This volume collects papers published in two Special Issues “Kinematics and Robot Design II, KaRD2019” (https://www.mdpi.com/journal/robotics/special_issues/KRD2019) and “Kinematics and Robot Design III, KaRD2020” (https://www.mdpi.com/journal/robotics/special_issues/KaRD2020), which are the second and third issues of the KaRD Special Issue series hosted by the open access journal robotics.The KaRD series is an open environment where researchers present their works and discuss all topics focused on the many aspects that involve kinematics in the design of robotic/automatic systems. It aims at being an established reference for researchers in the field as other serial international conferences/publications are. Even though the KaRD series publishes one Special Issue per year, all the received papers are peer-reviewed as soon as they are submitted and, if accepted, they are immediately published in MDPI Robotics. Kinematics is so intimately related to the design of robotic/automatic systems that the admitted topics of the KaRD series practically cover all the subjects normally present in well-established international conferences on “mechanisms and robotics”.KaRD2019 together with KaRD2020 received 22 papers and, after the peer-review process, accepted only 17 papers. The accepted papers cover problems related to theoretical/computational kinematics, to biomedical engineering and to other design/applicative aspects

13th Annual Review of Progress in Applied Computational Electromagnetics at the Naval Postgraduate School, Monterey, CA, March 17-21, 1997, Conference Proceedings Volumes I & II

Includes Volumes 1 &

Proceedings of the First International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics

1st International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Kruger Park, 8-10 April 2002.This lecture is a principle-based review of a growing body of fundamental work stimulated by multiple opportunities to optimize geometric form (shape, structure, configuration, rhythm, topology, architecture, geography) in systems for heat and fluid flow. Currents flow against resistances, and by generating entropy (irreversibility) they force the system global performance to levels lower than the theoretical limit. The system design is destined to remain imperfect because of constraints (finite sizes, costs, times). Improvements can be achieved by properly balancing the resistances, i.e., by spreading the imperfections through the system. Optimal spreading means to endow the system with geometric form. The system construction springs out of the constrained maximization of global performance. This 'constructal' design principle is reviewed by highlighting applications from heat transfer engineering. Several examples illustrate the optimized internal structure of convection cooled packages of electronics. The origin of optimal geometric features lies in the global effort to use every volume element to the maximum, i.e., to pack the element not only with the most heat generating components, but also with the most flow, in such a way that every fluid packet is effectively engaged in cooling. In flows that connect a point to a volume or an area, the resulting structure is a tree with high conductivity branches and low-conductivity interstices.tm201

Atti del XXXV Convegno Nazionale di Idraulica e Costruzioni Idrauliche

La XXXV edizione del Convegno Nazionale di Idraulica e Costruzioni Idrauliche (IDRA16), co-organizzata dal Gruppo Italiano di Idraulica (GII) e dal Dipartimento di Ingegneria Civile, Chimica, Ambientale, e dei Materiali (DICAM) dell’Alma Mater Studiorum - Università di Bologna, si è svolta a Bologna dal 14 al 16 settembre 2016. Il Convegno Nazionale è tornato pertanto ad affacciarsi all’ombra del “Nettuno”, dopo l’edizione del 1982 (XVIII edizione). Il titolo della XXXV edizione, “Ambiente, Risorse, Energia: le sfide dell’Ingegneria delle acque in un mondo che cambia”, sottolinea l’importanza e la complessità delle tematiche che rivestono la sfera dello studio e del governo delle risorse idriche. Le sempre più profonde interconnessioni tra risorse idriche, sviluppo economico e benessere sociale, infatti, spronano sia l’Accademia che l’intera comunità tecnico-scientifica nazionale ed internazionale all’identificazione ed alla messa in atto di strategie di gestione innovative ed ottimali: sfide percepite quanto mai necessarie in un contesto ambientale in continua evoluzione, come quello in cui viviamo. La XXXV edizione del Convegno di Idraulica e Costruzioni Idrauliche, pertanto, si è posta come punto d’incontro della comunità tecnico-scientifica italiana per la discussione a tutto tondo di tali problematiche, offrendo un programma scientifico particolarmente ricco e articolato, che ha coperto tutti gli ambiti riconducibili all’Ingegneria delle Acque. L’apertura dei lavori del Convegno si è svolta nella storica cornice della Chiesa di Santa Cristina, uno dei luoghi più caratteristici e belli della città ed oggi luogo privilegiato per l’ascolto della musica classica, mentre le attività di presentazione e discussione scientifica si sono svolte principalmente presso la sede della Scuola di Ingegneria e Architettura dell’Università di Bologna sita in Via Terracini. Il presente volume digitale ad accesso libero (licenza Creative Commons 4.0) raccoglie le memorie brevi pervenute al Comitato Scientifico di IDRA16 ed accettate per la presentazione al convegno a valle di un processo di revisione tra pari. Il volume articola dette memorie in sette macro-tematiche, che costituiscono i capitoli del volume stesso: I. meccanica dei fluidi; II. ambiente marittimo e costiero; III. criteri, metodi e modelli per l’analisi dei processi idrologici e la gestione delle acque; IV. gestione e tutela dei corpi idrici e degli ecosistemi; V. valutazione e mitigazione del rischio idrologico e idraulico; VI. dinamiche acqua-società: sviluppo sostenibile e gestione del territorio; VII. monitoraggio, open-data e software libero. Ciascuna macro-tematica raggruppa più sessioni specialistiche autonome sviluppatesi in parallelo durante le giornate del Convegno, i cui titoli vengono richiamati all’interno del presente volume. La vastità e la diversità delle tematiche affrontate, che ben rappresentano la complessità delle numerose sfide dell’Ingegneria delle Acque, appaiono evidenti dalla consultazione dell’insieme di memorie brevi presentate. La convinta partecipazione della Comunità Scientifica Italiana è dimostrata dalle oltre 350 memorie brevi, distribuite in maniera pressoché uniforme tra le sette macro-tematiche di riferimento. Dette memorie sono sommari estesi di lunghezza variabile redatti in lingua italiana, o inglese. In particolare, la possibilità di stesura in inglese è stata concessa con l’auspicio di portare la visibilità del lavoro presentato ad un livello sovranazionale, grazie alla pubblicazione open access del volume degli Atti del Convegno. Il volume si divide in tre parti: la parte iniziale è dedicata alla presentazione del volume ed all’indice generale dei contributi divisi per macro-tematiche; la parte centrale raccoglie le memorie brevi; la terza parte riporta l’indice analitico degli Autori, che chiude il volume

Safety and Reliability - Safe Societies in a Changing World

The contributions cover a wide range of methodologies and application areas for safety and reliability that contribute to safe societies in a changing world. These methodologies and applications include: - foundations of risk and reliability assessment and management - mathematical methods in reliability and safety - risk assessment - risk management - system reliability - uncertainty analysis - digitalization and big data - prognostics and system health management - occupational safety - accident and incident modeling - maintenance modeling and applications - simulation for safety and reliability analysis - dynamic risk and barrier management - organizational factors and safety culture - human factors and human reliability - resilience engineering - structural reliability - natural hazards - security - economic analysis in risk managemen