[PDF] Collocation And Galerkin Time Stepping Methods eBook

Author : National Aeronautics and Space Adm Nasa
Publisher : Independently Published
Page : 40 pages
File Size : 28,25 MB
Release : 2019-01-13
Category : Science
ISBN : 9781793962157

GET BOOK

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods. Huynh, H. T. Glenn Research Center NASA/TM-2011-216340, E-17277

Author : H. T. Huynh
Publisher : BiblioGov
Page : 42 pages
File Size : 28,87 MB
Release : 2013-06
Category :
ISBN : 9781289031084

GET BOOK

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.

Author : Hermann Brunner
Publisher : Cambridge University Press
Page : 620 pages
File Size : 39,56 MB
Release : 2004-11-15
Category : Mathematics
ISBN : 9780521806152

GET BOOK

Publisher Description

Author : John P. Boyd
Publisher : Courier Corporation
Page : 690 pages
File Size : 23,89 MB
Release : 2001-12-03
Category : Mathematics
ISBN : 0486411834

GET BOOK

Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.

Author : R. Loek Van Heyningen
Publisher :
Page : 69 pages
File Size : 22,22 MB
Release : 2021
Category :
ISBN :

GET BOOK

This thesis explores the ability of the discontinuous Galerkin (DG) method to numerically solve the Boltzmann equation. Constructing numerical methods for this equation is a challenge, due in part to the kinetic theory description of moving particles, which relies on space, time, and velocity variables. Two novel approaches are presented and compared. The first uses a spectral collocation basis in velocity space. The resulting system is solved in time using Diagonally Implicit Runge-Kutta methods, chosen in order to mitigate stiffness concerns. A Jacobian-Free Newton-Krylov method is presented, accelerated with a sweeping preconditioner. The method is tested on 1D and 2D problems in order to validate its convergence behavior and investigate its efficiency. The second method uses DG for moment equations, which can be derived as spectral methods in velocity space with spatial and temporal adaptivity. These methods were first proposed in 1949 by Grad, but their applicability has been limited. The equations are not guaranteed to be hyperbolic, leading to stability issues. The elegance and potential for cost-reduction of Grad's moment method have led to the development of different moment closures that preserve hyperbolicity and model accuracy. The approaches studied in this thesis, the globally hyperbolic moment methods, restore hyperbolicity by introducing a term that cannot be written in conservative form. The equations are typically solved with operator splitting and low-order methods. We examine the promise and challenges of applying a high-order DG method with explicit Runge-Kutta time-stepping to these equations on common 1D test cases. The thesis ends with a discussion on the prospects of both methods and suggestions for future work.

Author : David A. Kopriva
Publisher : Springer Science & Business Media
Page : 397 pages
File Size : 37,52 MB
Release : 2009-05-27
Category : Mathematics
ISBN : 9048122619

GET BOOK

This book explains how to solve partial differential equations numerically using single and multidomain spectral methods. It shows how only a few fundamental algorithms form the building blocks of any spectral code, even for problems with complex geometries.

Author : Hans Petter Langtangen
Publisher : Springer Nature
Page : 395 pages
File Size : 41,92 MB
Release : 2019-09-26
Category : Mathematics
ISBN : 3030237885

GET BOOK

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 : Dietrich Braess
Publisher : Cambridge University Press
Page : 348 pages
File Size : 41,41 MB
Release : 2007-04-12
Category : Mathematics
ISBN : 113946146X

GET BOOK

This definitive introduction to finite element methods was thoroughly updated for this 2007 third edition, which features important material for both research and application of the finite element method. The discussion of saddle-point problems is a highlight of the book and has been elaborated to include many more nonstandard applications. The chapter on applications in elasticity now contains a complete discussion of locking phenomena. The numerical solution of elliptic partial differential equations is an important application of finite elements and the author discusses this subject comprehensively. These equations are treated as variational problems for which the Sobolev spaces are the right framework. Graduate students who do not necessarily have any particular background in differential equations, but require an introduction to finite element methods will find this text invaluable. Specifically, the chapter on finite elements in solid mechanics provides a bridge between mathematics and engineering.

Author : Mass Per Pettersson
Publisher : Springer
Page : 217 pages
File Size : 33,38 MB
Release : 2015-03-10
Category : Technology & Engineering
ISBN : 3319107143

GET BOOK

This monograph presents computational techniques and numerical analysis to study conservation laws under uncertainty using the stochastic Galerkin formulation. With the continual growth of computer power, these methods are becoming increasingly popular as an alternative to more classical sampling-based techniques. The text takes advantage of stochastic Galerkin projections applied to the original conservation laws to produce a large system of modified partial differential equations, the solutions to which directly provide a full statistical characterization of the effect of uncertainties. Polynomial Chaos Methods of Hyperbolic Partial Differential Equations focuses on the analysis of stochastic Galerkin systems obtained for linear and non-linear convection-diffusion equations and for a systems of conservation laws; a detailed well-posedness and accuracy analysis is presented to enable the design of robust and stable numerical methods. The exposition is restricted to one spatial dimension and one uncertain parameter as its extension is conceptually straightforward. The numerical methods designed guarantee that the solutions to the uncertainty quantification systems will converge as the mesh size goes to zero. Examples from computational fluid dynamics are presented together with numerical methods suitable for the problem at hand: stable high-order finite-difference methods based on summation-by-parts operators for smooth problems, and robust shock-capturing methods for highly nonlinear problems. Academics and graduate students interested in computational fluid dynamics and uncertainty quantification will find this book of interest. Readers are expected to be familiar with the fundamentals of numerical analysis. Some background in stochastic methods is useful but notnecessary.