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Book High order finite difference approximations for hyperbolic problems

Download or read book High order finite difference approximations for hyperbolic problems written by Hannes Frenander and published by Linköping University Electronic Press. This book was released on 2017-01-24 with total page 54 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations. For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT). A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time. To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time. In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.

Book A Bibliography for the Numerical Solution of Partial Differential Equations

Download or read book A Bibliography for the Numerical Solution of Partial Differential Equations written by John H. Giese and published by . This book was released on 1969 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: A list of 2561 references to the numerical solution of partial differential equations has been compiled. References to reviews in several abstracting journals have been given, and a crude index has been prepared. (Author).

Book Essential Partial Differential Equations

Download or read book Essential Partial Differential Equations written by David F. Griffiths and published by Springer. This book was released on 2015-09-24 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these; the qualitative behaviour of solutions being established using classical concepts: maximum principles and energy methods. Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the approximation of multidimensional advection–diffusion problems. The underlying theory is illustrated by numerous examples and there are around 300 exercises, designed to promote and test understanding. They are starred according to level of difficulty. Solutions to odd-numbered exercises are available to all readers while even-numbered solutions are available to authorised instructors. Written in an informal yet rigorous style, Essential Partial Differential Equations is designed for mathematics undergraduates in their final or penultimate year of university study, but will be equally useful for students following other scientific and engineering disciplines in which PDEs are of practical importance. The only prerequisite is a familiarity with the basic concepts of calculus and linear algebra.

Book Exact Bounds for Solutions of Hyperbolic Equations by Finite Difference Methods

Download or read book Exact Bounds for Solutions of Hyperbolic Equations by Finite Difference Methods written by Hans F. Weinberger and published by . This book was released on 1959 with total page 32 pages. Available in PDF, EPUB and Kindle. Book excerpt: The mean square division of the finite difference approximation from the solution of the wave equation vanishing on the surfae of an N-cube and satisfying given Cauchy data is bound in terms of the Cauchy data. The bound is proportional to the mesh parameter h.

Book Numerical Approximation of Partial Differential Equations

Download or read book Numerical Approximation of Partial Differential Equations written by Alfio Quarteroni and published by Springer Science & Business Media. This book was released on 2008-09-24 with total page 550 pages. Available in PDF, EPUB and Kindle. Book excerpt: Everything is more simple than one thinks but at the same time more complex than one can understand Johann Wolfgang von Goethe To reach the point that is unknown to you, you must take the road that is unknown to you St. John of the Cross This is a book on the numerical approximation ofpartial differential equations (PDEs). Its scope is to provide a thorough illustration of numerical methods (especially those stemming from the variational formulation of PDEs), carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is our primary concern. Many kinds of problems are addressed: linear and nonlinear, steady and time-dependent, having either smooth or non-smooth solutions. Besides model equations, we consider a number of (initial-) boundary value problems of interest in several fields of applications. Part I is devoted to the description and analysis of general numerical methods for the discretization of partial differential equations. A comprehensive theory of Galerkin methods and its variants (Petrov Galerkin and generalized Galerkin), as wellas ofcollocationmethods, is devel oped for the spatial discretization. This theory is then specified to two numer ical subspace realizations of remarkable interest: the finite element method (conforming, non-conforming, mixed, hybrid) and the spectral method (Leg endre and Chebyshev expansion).

Book Advanced Numerical Approximation of Nonlinear Hyperbolic Equations

Download or read book Advanced Numerical Approximation of Nonlinear Hyperbolic Equations written by B. Cockburn and published by C.I.M.E. Foundation Subseries. This book was released on 1998-11-18 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the texts of the four series of lectures presented by B.Cockburn, C.Johnson, C.W. Shu and E.Tadmor at a C.I.M.E. Summer School. It is aimed at providing a comprehensive and up-to-date presentation of numerical methods which are nowadays used to solve nonlinear partial differential equations of hyperbolic type, developing shock discontinuities. The most effective methodologies in the framework of finite elements, finite differences, finite volumes spectral methods and kinetic methods, are addressed, in particular high-order shock capturing techniques, discontinuous Galerkin methods, adaptive techniques based upon a-posteriori error analysis.

Book Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

Download or read book Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems written by Sergey I. Kabanikhin and published by Walter de Gruyter. This book was released on 2013-04-09 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider dynamic types of inverse problems in which the additional information is given by the trace of the direct problem on a (usually time-like) surface of the domain. They discuss theoretical and numerical background of the finite-difference scheme inversion, the linearization method, the method of Gel'fand-Levitan-Krein, the boundary control method, and the projection method and prove theorems of convergence, conditional stability, and other properties of the mentioned methods.

Book Bounds on the Truncation Error by Finite Differences for the Goursat Problem

Download or read book Bounds on the Truncation Error by Finite Differences for the Goursat Problem written by Abdul Kadir Aziz and published by . This book was released on 1962 with total page 38 pages. Available in PDF, EPUB and Kindle. Book excerpt: A finite difference analogue is formulated for the boundary value problem, a finite analogue of Riemann's function is developed. It is shown that the truncation error is bounded explicitly in terms of the data of the problem, which is of 0(h squared), where h is the mesh size. Bounds of the same order are obtained for the error in approximating the derivatives. Similar results are derived for the nonlinear cas if f satisfies certain continuity and differentiability conditions.

Book Finite Difference Approximations for Hyperbolic Systems with Two Boundaries

Download or read book Finite Difference Approximations for Hyperbolic Systems with Two Boundaries written by Meegyeong Paik and published by . This book was released on 1995 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Numerical Solution of Differential Equations

Download or read book Numerical Solution of Differential Equations written by Zhilin Li and published by Cambridge University Press. This book was released on 2017-11-30 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: A practical and concise guide to finite difference and finite element methods. Well-tested MATLAB® codes are available online.

Book U S Government Research Reports

Download or read book U S Government Research Reports written by and published by . This book was released on 1963 with total page 754 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Acta Mathematica Academiae Scientiarum Hungaricae

Download or read book Acta Mathematica Academiae Scientiarum Hungaricae written by and published by . This book was released on 1963 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Partial Differential Equations with Numerical Methods

Download or read book Partial Differential Equations with Numerical Methods written by Stig Larsson and published by Springer Science & Business Media. This book was released on 2008-12-05 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.