Collocation Methods For Parabolic Partial Differential Equations In One Space Dimension
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Author | : John H. Cerutti |
Publisher | : |
Total Pages | : 30 |
Release | : 1975 |
Genre | : |
ISBN | : |
Collocation at Gaussian points for a scalar m'th order ordinary differential equation has been studied by C. de Boor and B. Swartz. J. Douglas, Jr. and T. Dupont, using collocation at Gaussian points, and a combination of energy estimates and approximation theory have given a comprehensive theory for parabolic problems in a single space variable. While the results of this report parallel those of Douglas and Dupont, the approach is basically different. The Laplace transform is used to 'lift' the results of de Boor and Swartz to linear parabolic problems. This indicates a general procedure that may be used to 'lift' schemes for elliptic problems to schemes for parabolic problems. Additionally there is a section on longtime integration and A-stability.
Author | : J.jr. Douglas |
Publisher | : |
Total Pages | : 162 |
Release | : 1974-05-28 |
Genre | : Mathematics |
ISBN | : |
Author | : Qiang Zheng |
Publisher | : |
Total Pages | : 0 |
Release | : 2005 |
Genre | : |
ISBN | : |
This thesis presents a new class of collocation methods for the approximate numerical solution of linear parabolic partial differential equations. In the time dimension, the partial derivative with respect to time is replaced by finite differences, to form the implicit Euler method. At each time step, a polynomial approximating the exact solution is calculated for each triangular finite element created by the Rivara algorithm. Polynomials of adjacent finite elements have matching values and matching normal derivatives at a set of discrete points, called "matching points". The method of nested dissection is used to eliminate all variables at the interior matching points of the domain. The maximum error of the solution is of the order of the time step size, which is O (dt), except when dt is sufficiently small. In that case, the maximum error can be very small, depending on the density of the space mesh. An application based on OpenGL and Motif to visualize the solutions is also described in this thesis. Extensive numerical results, pictures of refined meshes, and 3 D representations of the solutions are given.
Author | : J.jr. Douglas |
Publisher | : |
Total Pages | : 160 |
Release | : 2014-01-15 |
Genre | : |
ISBN | : 9783662203378 |
Author | : Jacques Tits |
Publisher | : |
Total Pages | : 299 |
Release | : 1974 |
Genre | : Collocation methods |
ISBN | : 9780387067476 |
Author | : William E. Schiesser |
Publisher | : John Wiley & Sons |
Total Pages | : 566 |
Release | : 2017-05-22 |
Genre | : Mathematics |
ISBN | : 1119301033 |
A comprehensive approach to numerical partial differential equations Spline Collocation Methods for Partial Differential Equations combines the collocation analysis of partial differential equations (PDEs) with the method of lines (MOL) in order to simplify the solution process. Using a series of example applications, the author delineates the main features of the approach in detail, including an established mathematical framework. The book also clearly demonstrates that spline collocation can offer a comprehensive method for numerical integration of PDEs when it is used with the MOL in which spatial (boundary value) derivatives are approximated with splines, including the boundary conditions. R, an open-source scientific programming system, is used throughout for programming the PDEs and numerical algorithms, and each section of code is clearly explained. As a result, readers gain a complete picture of the model and its computer implementation without having to fill in the details of the numerical analysis, algorithms, or programming. The presentation is not heavily mathematical, and in place of theorems and proofs, detailed example applications are provided. Appropriate for scientists, engineers, and applied mathematicians, Spline Collocation Methods for Partial Differential Equations: Introduces numerical methods by first presenting basic examples followed by more complicated applications Employs R to illustrate accurate and efficient solutions of the PDE models Presents spline collocation as a comprehensive approach to the numerical integration of PDEs and an effective alternative to other, well established methods Discusses how to reproduce and extend the presented numerical solutions Identifies the use of selected algorithms, such as the solution of nonlinear equations and banded or sparse matrix processing Features a companion website that provides the related R routines Spline Collocation Methods for Partial Differential Equations is a valuable reference and/or self-study guide for academics, researchers, and practitioners in applied mathematics and engineering, as well as for advanced undergraduates and graduate-level students.
Author | : Leon Lapidus |
Publisher | : John Wiley & Sons |
Total Pages | : 677 |
Release | : 2011-02-14 |
Genre | : Mathematics |
ISBN | : 1118031210 |
From the reviews of Numerical Solution of PartialDifferential Equations in Science and Engineering: "The book by Lapidus and Pinder is a very comprehensive, evenexhaustive, survey of the subject . . . [It] is unique in that itcovers equally finite difference and finite element methods." Burrelle's "The authors have selected an elementary (but not simplistic)mode of presentation. Many different computational schemes aredescribed in great detail . . . Numerous practical examples andapplications are described from beginning to the end, often withcalculated results given." Mathematics of Computing "This volume . . . devotes its considerable number of pages tolucid developments of the methods [for solving partial differentialequations] . . . the writing is very polished and I found it apleasure to read!" Mathematics of Computation Of related interest . . . NUMERICAL ANALYSIS FOR APPLIED SCIENCE Myron B. Allen andEli L. Isaacson. A modern, practical look at numerical analysis,this book guides readers through a broad selection of numericalmethods, implementation, and basic theoretical results, with anemphasis on methods used in scientific computation involvingdifferential equations. 1997 (0-471-55266-6) 512 pp. APPLIED MATHEMATICS Second Edition, J. David Logan.Presenting an easily accessible treatment of mathematical methodsfor scientists and engineers, this acclaimed work covers fluidmechanics and calculus of variations as well as more modernmethods-dimensional analysis and scaling, nonlinear wavepropagation, bifurcation, and singular perturbation. 1996(0-471-16513-1) 496 pp.
Author | : Rong Wang |
Publisher | : |
Total Pages | : 0 |
Release | : 2002 |
Genre | : |
ISBN | : |
Author | : John Howell Cerutti |
Publisher | : |
Total Pages | : 260 |
Release | : 1975 |
Genre | : |
ISBN | : |
Author | : Ulrich Langer |
Publisher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 261 |
Release | : 2019-09-23 |
Genre | : Mathematics |
ISBN | : 3110548488 |
This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.