Collocation Methods For Linear Parabolic Partial Differential Equations
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| 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 | : 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 | : John Howell Cerutti |
| Publisher | : |
| Total Pages | : 260 |
| Release | : 1975 |
| Genre | : |
| ISBN | : |
| Author | : J.jr. Douglas |
| Publisher | : |
| Total Pages | : 162 |
| Release | : 1974-05-28 |
| Genre | : Mathematics |
| ISBN | : |
| Author | : Nicola Bellomo |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 206 |
| Release | : 2007-09-26 |
| Genre | : Mathematics |
| ISBN | : 0817646108 |
Analysis of nonlinear models and problems is crucial in the application of mathematics to real-world problems. This book approaches this important topic by focusing on collocation methods for solving nonlinear evolution equations and applying them to a variety of mathematical problems. These include wave motion models, hydrodynamic models of vehicular traffic flow, convection-diffusion models, reaction-diffusion models, and population dynamics models. The book may be used as a textbook for graduate courses on collocation methods, nonlinear modeling, and nonlinear differential equations. Examples and exercises are included in every chapter.
| Author | : J.jr. Douglas |
| Publisher | : |
| Total Pages | : 160 |
| Release | : 2014-01-15 |
| Genre | : |
| ISBN | : 9783662203378 |
| Author | : William F. Ames |
| Publisher | : Academic Press |
| Total Pages | : 380 |
| Release | : 2014-05-10 |
| Genre | : Mathematics |
| ISBN | : 1483262421 |
Numerical Methods for Partial Differential Equations, Second Edition deals with the use of numerical methods to solve partial differential equations. In addition to numerical fluid mechanics, hopscotch and other explicit-implicit methods are also considered, along with Monte Carlo techniques, lines, fast Fourier transform, and fractional steps methods. Comprised of six chapters, this volume begins with an introduction to numerical calculation, paying particular attention to the classification of equations and physical problems, asymptotics, discrete methods, and dimensionless forms. Subsequent chapters focus on parabolic and hyperbolic equations, elliptic equations, and special topics ranging from singularities and shocks to Navier-Stokes equations and Monte Carlo methods. The final chapter discuss the general concepts of weighted residuals, with emphasis on orthogonal collocation and the Bubnov-Galerkin method. The latter procedure is used to introduce finite elements. This book should be a valuable resource for students and practitioners in the fields of computer science and applied mathematics.
| Author | : Motsa Sandile Sydney |
| Publisher | : |
| Total Pages | : |
| Release | : 2016 |
| Genre | : Mathematics |
| ISBN | : |
In this study we introduce the multidomain bivariate spectral collocation method for solving nonlinear parabolic partial differential equations (PDEs) that are defined over large time intervals. The main idea is to reduce the size of the computational domain at each subinterval to ensure that very accurate results are obtained within shorter computational time when the spectral collocation method is applied. The proposed method is based on applying the quasi-linearization technique to simplify the nonlinear partial differential equation (PDE) first. The time domain is decomposed into smaller nonoverlapping subintervals. Discretization is then performed on both time and space variables using spectral collocation. The approximate solution of the PDE is obtained by solving the resulting linear matrix system at each subinterval independently. When the solution in the first subinterval has been computed, the continuity condition is used to obtain the initial guess in subsequent subintervals. The solutions at different subintervals are matched together along a common boundary. The examples chosen for numerical experimentation include the Burger's-Fisher equation, the Fitzhugh-Nagumo equation and the Burger's-Huxley equation. To demonstrate the accuracy and the effectiveness of the proposed method, the computational time and the error analysis of the chosen illustrative examples are presented in the tables.
| Author | : Yong Wang |
| Publisher | : National Library of Canada = Bibliothèque nationale du Canada |
| Total Pages | : |
| Release | : 1994 |
| Genre | : |
| ISBN | : 9780315989535 |
| 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.
