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Author: Viktor Linders
Publisher: Linköping University Electronic Press
ISBN: 9176854272
Category :
Languages : en
Pages : 44
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Book Description
In this thesis we consider errors arising from finite difference operators on summation-by-parts (SBP) form, used in the discretisation of partial differential equations. The SBP operators are augmented with simultaneous-approximation-terms (SATs) to weakly impose boundary conditions. The SBP-SAT framework combines high order of accuracy with a systematic construction of provably stable boundary procedures, which renders it suitable for a wide range of problems. The first part of the thesis treats wave propagation problems discretised using SBP operators on coarse grids. Unless special care is taken, inaccurate approximations of the underlying dispersion relation materialises in the form of an incorrect propagation speed. We present a procedure for constructing SBP operators with minimal dispersion error. Experiments indicate that they outperform higher order non-optimal SBP operators for flow problems involving high frequencies and long simulation times. In the second part of the thesis, the formal order of accuracy of SBP operators near boundaries is analysed. We prove that the order in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. This generalises the classical theory posed on uniform and conforming grids. We further show that for a common class of SBP operators, the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid. In the final contribution if the thesis, we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability analyses are performed for continuous and discrete problems. A general condition is obtained that is necessary and sufficient for the transmission problem to satisfy an energy estimate. The theory provides insights into the coupling of fluid flow models, multi-block formulations, numerical filters, interpolation and multi-grid implementations.
Author: Viktor Linders
Publisher: Linköping University Electronic Press
ISBN: 9176854272
Category :
Languages : en
Pages : 44
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Book Description
In this thesis we consider errors arising from finite difference operators on summation-by-parts (SBP) form, used in the discretisation of partial differential equations. The SBP operators are augmented with simultaneous-approximation-terms (SATs) to weakly impose boundary conditions. The SBP-SAT framework combines high order of accuracy with a systematic construction of provably stable boundary procedures, which renders it suitable for a wide range of problems. The first part of the thesis treats wave propagation problems discretised using SBP operators on coarse grids. Unless special care is taken, inaccurate approximations of the underlying dispersion relation materialises in the form of an incorrect propagation speed. We present a procedure for constructing SBP operators with minimal dispersion error. Experiments indicate that they outperform higher order non-optimal SBP operators for flow problems involving high frequencies and long simulation times. In the second part of the thesis, the formal order of accuracy of SBP operators near boundaries is analysed. We prove that the order in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. This generalises the classical theory posed on uniform and conforming grids. We further show that for a common class of SBP operators, the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid. In the final contribution if the thesis, we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability analyses are performed for continuous and discrete problems. A general condition is obtained that is necessary and sufficient for the transmission problem to satisfy an energy estimate. The theory provides insights into the coupling of fluid flow models, multi-block formulations, numerical filters, interpolation and multi-grid implementations.
Author: Fatemeh Ghasemi Zinatabadi
Publisher: Linköping University Electronic Press
ISBN: 9176850315
Category :
Languages : en
Pages : 44
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Book Description
In this thesis, we consider the numerical solution of initial boundary value problems (IBVPs). Boundary and interface conditions are derived such that the IBVP under consideration is well-posed. We also study the dual problem and the related dual boundary/interface conditions. Once the continuous problem is analyzed, we use finite difference operators with the Summation- By-Parts property (SBP) and a weak boundary/interface treatment using the Simultaneous-Approximation-Terms (SAT) technique to construct high-order accurate numerical schemes. We focus in particular on stability, conservation and dual consistency. The energy method is used as our main analysis tool for both the continuous and numerical problems. The contributions of this thesis can be divided into two parts. The first part focuses on the coupling of different IBVPs. Interface conditions are derived such that the continuous problem satisfy an energy estimate and such that the discrete problem is stable. In the first paper, two hyperbolic systems of different size posed on two domains are considered. We derive the dual problem and dual interface conditions. It is also shown that a specific choice of penalty matrices leads to dual consistency. As an application, we study the coupling of the Euler and wave equations. In the fourth paper, we examine how to couple the compressible and incompressible Navier-Stokes equations. In order to obtain a sufficient number of interface conditions, the decoupled heat equation is added to the incompressible equations. The interface conditions include mass and momentum balance and two variants of heat transfer. The typical application in this case is the atmosphere-ocean coupling. The second part of the thesis focuses on the relation between the primal and dual problem and the relation between dual consistency and conservation. In the second and third paper, we show that dual consistency and conservation are equivalent concepts for linear hyperbolic conservation laws. We also show that these concepts are equivalent for symmetric or symmetrizable parabolic problems in the fifth contribution. The relation between the primal and dual boundary conditions for linear hyperbolic systems of equations is investigated in the sixth and last paper. It is shown that for given well-posed primal/dual boundary conditions, the corresponding well-posed dual/primal boundary conditions can be obtained by a simple scaling operation. It is also shown how one can proceed directly from the well-posed weak primal problem to the well-posed weak dual problem. Den här avhandlingen handlar om numeriska metoder för att lösa initial och randvärdes problem. Studien fokuserar på härledningen av rand/kopplingsvillkor som garanterar välställdhet. Det duala problemet och dess duala rand/kopplingsvillkor studeras också. Dessa problem diskretiseras genom att använda noggranna finita differensscheman på SBP-form (eng. summation-by-parts), kombinerat med en svag randbehandling benämnd SAT (eng. simultaneous approximation term). Vi fokuserar särskilt på stabilitet, konservation och dualkonsistens. Det främsta analysverktyget för både det kontinuerliga och diskreta problemet är energimetoden. Den första delen av avhandlingen behandlar välställdhet och stabilitet för koppling av olika system av ekvationer. Kopplingsvillkoren är härledda så att det kontinuerliga problemet uppfyller en energiuppskattning och så att det diskreta problemet är stabilt. I den första artikeln görs analysen för koppling av två olika hyperboliska system på första ordningens form. Som tillämpning kopplar vi Euler och vågekvationerna. Koppling mellan kompressibla och inkompressibla Navier-Stokes ekvationer studeras i den fjärde artikeln. För att få rätt antal kopplingsvillkor lägger vi till värmeledningsekvationen till de inkompressibla ekvationerna. Kopplingsvillkoren innefattar massans och rörelsemängdens bevarande samt två varianter av värmeöverföring. Den typiska tillämpningen är koppling mellan atmosfär och hav. Den andra delen av avhandlingen fokuserar på relationen mellan det primära och duala problemet och relationen mellan dualkonsistens och konservation. I den andra och tredje artikeln visar vi att dualkonsistens och konservation är ekvivalenta koncept för linjära hyperboliska konserveringslagar. I den femte artikeln, visas att dessa koncept är ekvivalenta även för paraboliska problem. Relationen mellan de primära och duala randvilkoren för linjära hyperboliska system av ekvationer i två dimensioner studeras i den sista artikeln. Vi visar att primära/duala välställda randvilkor ger duala/primära välställda randvilkor genom en enkel skalningsoperation. Det visas också att man kan gå direkt från det välställda svaga primära problemet till det välställda svaga duala problemet.
Author: Andrea Alessandro Ruggiu
Publisher: Linköping University Electronic Press
ISBN: 9176850234
Category :
Languages : en
Pages : 57
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Book Description
Many physical phenomena can be described mathematically by means of partial differential equations. These mathematical formulations are said to be well-posed if a unique solution, bounded by the given data, exists. The boundedness of the solution can be established through the so-called energy-method, which leads to an estimate of the solution by means of integration-by-parts. Numerical approximations mimicking integration-by-parts discretely are said to fulfill the Summation-By-Parts (SBP) property. These formulations naturally yield bounded approximate solutions if the boundary conditions are weakly imposed through Simultaneous-Approximation-Terms (SAT). Discrete problems with bounded solutions are said to be energy-stable. Energy-stable and high-order accurate SBP-SAT discretizations for well-posed linear problems were first introduced for centered finite-difference methods. These mathematical formulations, based on boundary conforming grids, allow for an exact mimicking of integration-by-parts. However, other discretizations techniques that do not include one or both boundary nodes, such as pseudo-spectral collocation methods, only fulfill a generalized SBP (GSBP) property but still lead to energy-stable solutions. This thesis consists of two main topics. The first part, which is mostly devoted to theoretical investigations, treats discretizations based on SBP and GSBP operators. A numerical approximation of a conservation law is said to be conservative if the approximate solution mimics the physical conservation property. It is shown that conservative and energy-stable spatial discretizations of variable coefficient problems require an exact numerical mimicking of integration-by-parts. We also discuss the invertibility of the algebraic problems arising from (G)SBP-SAT discretizations in time of energy-stable spatial approximations. We prove that pseudo-spectral collocation methods for the time derivative lead to invertible fully-discrete problems. The same result is proved for second-, fourth- and sixth-order accurate finite-difference based time integration methods. Once the invertibility of (G)SBP-SAT discrete formulations is established, we are interested in efficient algorithms for the unique solution of such problems. To this end, the second part of the thesis has a stronger experimental flavour and deals with convergence acceleration techniques for SBP-SAT approximations. First, we consider a modified Dual Time-Stepping (DTS) technique which makes use of two derivatives in pseudo-time. The new DTS formulation, compared to the classical one, accelerates the convergence to steady-state and reduces the stiffness of the problem. Next, we investigate multi-grid methods. For parabolic problems, highly oscillating error modes are optimally damped by iterative methods, while smooth residuals are transferred to coarser grids. In this case, we show that the Galerkin condition in combination with the SBP-preserving interpolation operators leads to fast convergence. For hyperbolic problems, low frequency error modes are rapidly expelled by grid coarsening, since coarser grids have milder stability restrictions on time steps. For such problems, Total Variation Dimishing Multi-Grid (TVD-MG) allows for faster wave propagation of first order upwind discretizations. In this thesis, we extend low order TVD-MG schemes to high-order SBP-SAT upwind discretizations.
Author: Peter Moczo
Publisher: Cambridge University Press
ISBN: 1139867695
Category : Science
Languages : en
Pages : 387
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Book Description
Among all the numerical methods in seismology, the finite-difference (FD) technique provides the best balance of accuracy and computational efficiency. This book offers a comprehensive introduction to FD and its applications to earthquake motion. Using a systematic tutorial approach, the book requires only undergraduate degree-level mathematics and provides a user-friendly explanation of the relevant theory. It explains FD schemes for solving wave equations and elastodynamic equations of motion in heterogeneous media, and provides an introduction to the rheology of viscoelastic and elastoplastic media. It also presents an advanced FD time-domain method for efficient numerical simulations of earthquake ground motion in realistic complex models of local surface sedimentary structures. Accompanied by a suite of online resources to help put the theory into practice, this is a vital resource for professionals and academic researchers using numerical seismological techniques, and graduate students in earthquake seismology, computational and numerical modelling, and applied mathematics.
Author: Justin Solomon
Publisher: CRC Press
ISBN: 1482251892
Category : Computers
Languages : en
Pages : 400
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Book Description
Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics presents a new approach to numerical analysis for modern computer scientists. Using examples from a broad base of computational tasks, including data processing, computational photography, and animation, the textbook introduces numerical modeling and algorithmic desig
Author: F. B. Hildebrand
Publisher: Courier Corporation
ISBN: 0486318559
Category : Mathematics
Languages : en
Pages : 708
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Book Description
Well-known, respected introduction, updated to integrate concepts and procedures associated with computers. Computation, approximation, interpolation, numerical differentiation and integration, smoothing of data, more. Includes 150 additional problems in this edition.
Author: Vladimir A. Dobrushkin
Publisher: CRC Press
ISBN: 142006830X
Category : Computers
Languages : en
Pages : 804
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Book Description
Explores the Impact of the Analysis of Algorithms on Many Areas within and beyond Computer Science A flexible, interactive teaching format enhanced by a large selection of examples and exercises Developed from the author’s own graduate-level course, Methods in Algorithmic Analysis presents numerous theories, techniques, and methods used for analyzing algorithms. It exposes students to mathematical techniques and methods that are practical and relevant to theoretical aspects of computer science. After introducing basic mathematical and combinatorial methods, the text focuses on various aspects of probability, including finite sets, random variables, distributions, Bayes’ theorem, and Chebyshev inequality. It explores the role of recurrences in computer science, numerical analysis, engineering, and discrete mathematics applications. The author then describes the powerful tool of generating functions, which is demonstrated in enumeration problems, such as probabilistic algorithms, compositions and partitions of integers, and shuffling. He also discusses the symbolic method, the principle of inclusion and exclusion, and its applications. The book goes on to show how strings can be manipulated and counted, how the finite state machine and Markov chains can help solve probabilistic and combinatorial problems, how to derive asymptotic results, and how convergence and singularities play leading roles in deducing asymptotic information from generating functions. The final chapter presents the definitions and properties of the mathematical infrastructure needed to accommodate generating functions. Accompanied by more than 1,000 examples and exercises, this comprehensive, classroom-tested text develops students’ understanding of the mathematical methodology behind the analysis of algorithms. It emphasizes the important relation between continuous (classical) mathematics and discrete mathematics, which is the basis of computer science.
Author: Hans Petter Langtangen
Publisher: Springer Nature
ISBN: 3030237885
Category : Mathematics
Languages : en
Pages : 405
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Book Description
This textbook teaches finite element methods from a computational point of view. It focuses on how to develop flexible computer programs with Python, a programming language in which a combination of symbolic and numerical tools is used to achieve an explicit and practical derivation of finite element algorithms. The finite element library FEniCS is used throughout the book, but the content is provided in sufficient detail to ensure that students with less mathematical background or mixed programming-language experience will equally benefit. All program examples are available on the Internet.
Author: Remi Abgrall
Publisher: Elsevier
ISBN: 0444637958
Category : Mathematics
Languages : en
Pages : 668
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Book Description
Handbook of Numerical Methods for Hyperbolic Problems explores the changes that have taken place in the past few decades regarding literature in the design, analysis and application of various numerical algorithms for solving hyperbolic equations. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and readily understand their relative advantages and limitations. - Provides detailed, cutting-edge background explanations of existing algorithms and their analysis - Ideal for readers working on the theoretical aspects of algorithm development and its numerical analysis - Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or readers involved in applications - Written by leading subject experts in each field who provide breadth and depth of content coverage
Author: Mark A. Pinsky
Publisher: American Mathematical Soc.
ISBN: 0821868896
Category : Mathematics
Languages : en
Pages : 545
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Book Description
Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.
