[PDF] Method Of Lines Approach To The Numerical Solution Of Conservation Laws eBook

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Page : pages
File Size : 13,40 MB
Release : 1979
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New explicit finite difference methods are developed for approximating the discontinuous time dependent solutions of nonlinear hyperbolic conservation laws. The analysis is based on the method of lines approach of decoupling the space and time discretizations and analyzing each independently before combining them into a composite method. Particular attention is given analyzing to high order spatial differences, artificial dissipation and the accurate approximation of boundary conditions. Both a third order iterated leap-frog predictor-corrector and a second order iterated Runge--Kutta method are shown to have excellent stability and accuracy properties for the time integration. These methods are A-stable when iterated to convergence and have the special property of allowing for local improvements in the stability and accuracy of the computed solution. The paper is designed to aid a scientist or engineer construct a numerical method specially tailored to a specific problem. The analysis requires an elementary knowledge of the numerical solution of ordinary differential equations, finite difference theory and gas dynamics.

Author : LEVEQUE
Publisher : Birkhäuser
Page : 221 pages
File Size : 28,70 MB
Release : 2013-11-11
Category : Science
ISBN : 3034851162

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These notes developed from a course on the numerical solution of conservation laws first taught at the University of Washington in the fall of 1988 and then at ETH during the following spring. The overall emphasis is on studying the mathematical tools that are essential in de veloping, analyzing, and successfully using numerical methods for nonlinear systems of conservation laws, particularly for problems involving shock waves. A reasonable un derstanding of the mathematical structure of these equations and their solutions is first required, and Part I of these notes deals with this theory. Part II deals more directly with numerical methods, again with the emphasis on general tools that are of broad use. I have stressed the underlying ideas used in various classes of methods rather than present ing the most sophisticated methods in great detail. My aim was to provide a sufficient background that students could then approach the current research literature with the necessary tools and understanding. vVithout the wonders of TeX and LaTeX, these notes would never have been put together. The professional-looking results perhaps obscure the fact that these are indeed lecture notes. Some sections have been reworked several times by now, but others are still preliminary. I can only hope that the errors are not too blatant. Moreover, the breadth and depth of coverage was limited by the length of these courses, and some parts are rather sketchy.

Author : Jan S. Hesthaven
Publisher : SIAM
Page : 571 pages
File Size : 21,13 MB
Release : 2018-01-30
Category : Science
ISBN : 1611975107

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Conservation laws are the mathematical expression of the principles of conservation and provide effective and accurate predictive models of our physical world. Although intense research activity during the last decades has led to substantial advances in the development of powerful computational methods for conservation laws, their solution remains a challenge and many questions are left open; thus it is an active and fruitful area of research. Numerical Methods for Conservation Laws: From Analysis to Algorithms offers the first comprehensive introduction to modern computational methods and their analysis for hyperbolic conservation laws, building on intense research activities for more than four decades of development; discusses classic results on monotone and finite difference/finite volume schemes, but emphasizes the successful development of high-order accurate methods for hyperbolic conservation laws; addresses modern concepts of TVD and entropy stability, strongly stable Runge-Kutta schemes, and limiter-based methods before discussing essentially nonoscillatory schemes, discontinuous Galerkin methods, and spectral methods; explores algorithmic aspects of these methods, emphasizing one- and two-dimensional problems and the development and analysis of an extensive range of methods; includes MATLAB software with which all main methods and computational results in the book can be reproduced; and demonstrates the performance of many methods on a set of benchmark problems to allow direct comparisons. Code and other supplemental material will be available online at publication.

Author : Jan S. Hesthaven
Publisher : SIAM
Page : 571 pages
File Size : 24,71 MB
Release : 2018-01-30
Category : Science
ISBN : 1611975093

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Conservation laws are the mathematical expression of the principles of conservation and provide effective and accurate predictive models of our physical world. Although intense research activity during the last decades has led to substantial advances in the development of powerful computational methods for conservation laws, their solution remains a challenge and many questions are left open; thus it is an active and fruitful area of research. Numerical Methods for Conservation Laws: From Analysis to Algorithms: offers the first comprehensive introduction to modern computational methods and their analysis for hyperbolic conservation laws, building on intense research activities for more than four decades of development; discusses classic results on monotone and finite difference/finite volume schemes, but emphasizes the successful development of high-order accurate methods for hyperbolic conservation laws; addresses modern concepts of TVD and entropy stability, strongly stable Runge-Kutta schemes, and limiter-based methods before discussing essentially nonoscillatory schemes, discontinuous Galerkin methods, and spectral methods; explores algorithmic aspects of these methods, emphasizing one- and two-dimensional problems and the development and analysis of an extensive range of methods; includes MATLAB software with which all main methods and computational results in the book can be reproduced; and demonstrates the performance of many methods on a set of benchmark problems to allow direct comparisons. Code and other supplemental material are available online at www.siam.org/books/cs18.

Author : R. Vichnevetsky
Publisher : SIAM
Page : 146 pages
File Size : 23,1 MB
Release : 1982-01-01
Category : Technology & Engineering
ISBN : 0898713927

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This book provides useful reference material for those concerned with the use of Fourier analysis and computational fluid dynamics.

Author : Bengt Fornberg
Publisher : Cambridge University Press
Page : 248 pages
File Size : 19,33 MB
Release : 1998-10-28
Category : Mathematics
ISBN : 9780521645645

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This book explains how, when and why the pseudospectral approach works.

Author : G. Evans
Publisher : Springer Science & Business Media
Page : 308 pages
File Size : 30,19 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 1447103793

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This is the practical introduction to the analytical approach taken in Volume 2. Based upon courses in partial differential equations over the last two decades, the text covers the classic canonical equations, with the method of separation of variables introduced at an early stage. The characteristic method for first order equations acts as an introduction to the classification of second order quasi-linear problems by characteristics. Attention then moves to different co-ordinate systems, primarily those with cylindrical or spherical symmetry. Hence a discussion of special functions arises quite naturally, and in each case the major properties are derived. The next section deals with the use of integral transforms and extensive methods for inverting them, and concludes with links to the use of Fourier series.