Download or read book Discontinuous Galerkin Finite Element Method for the Nonlinear Hyperbolic Problems with Entropy based Artificial Viscosity Stabilization written by Valentin Zingan and published by . This book was released on 2012 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This work develops a discontinuous Galerkin finite element discretization of non- linear hyperbolic conservation equations with efficient and robust high order stabilization built on an entropy-based artificial viscosity approximation. The solutions of equations are represented by elementwise polynomials of an arbitrary degree p> 0 which are continuous within each element but discontinuous on the boundaries. The discretization of equations in time is done by means of high order explicit Runge-Kutta methods identified with respective Butcher tableaux. To stabilize a numerical solution in the vicinity of shock waves and simultaneously preserve the smooth parts from smearing, we add some reasonable amount of artificial viscosity in accordance with the physical principle of entropy production in the interior of shock waves. The viscosity coefficient is proportional to the local size of the residual of an entropy equation and is bounded from above by the first-order artificial viscosity defined by a local wave speed. Since the residual of an entropy equation is supposed to be vanishingly small in smooth regions (of the order of the Local Truncation Error) and arbitrarily large in shocks, the entropy viscosity is almost zero everywhere except the shocks, where it reaches the first-order upper bound. One- and two-dimensional benchmark test cases are presented for nonlinear hyperbolic scalar conservation laws and the system of compressible Euler equations. These tests demonstrate the satisfactory stability properties of the method and optimal convergence rates as well. All numerical solutions to the test problems agree well with the reference solutions found in the literature. We conclude that the new method developed in the present work is a valuable alternative to currently existing techniques of viscous stabilization.
Download or read book Discontinuous Galerkin Finite Element Methods with Shock capturing for Nonlinear Convection Dominated Models written by Hamdullah Yücel and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: In this paper, convection-diffusion-reaction models with nonlinear reaction mechanisms, which are typical problems of chemical systems, are studied by using the upwind symmetric interior penalty Galerkin (SIPG) method. The local spurious oscillations are minimized by adding an artificial viscosity diffusion term to the original equations. A discontinuity sensor is used to detect the layers where unphysical oscillations occur. Finally, the proposed method is tested on various single- and multi-component problems.
Download or read book Handbook of Numerical Methods for Hyperbolic Problems written by Remi Abgrall and published by Elsevier. This book was released on 2016-11-17 with total page 668 pages. Available in PDF, EPUB and Kindle. Book excerpt: Handbook of Numerical Methods for Hyperbolic Problems explores the changes that have taken place in the past few decades regarding literature in the design, analysis and application of various numerical algorithms for solving hyperbolic equations. This volume provides concise summaries from experts in different types of algorithms, so that readers can find a variety of algorithms under different situations and readily understand their relative advantages and limitations. - Provides detailed, cutting-edge background explanations of existing algorithms and their analysis - Ideal for readers working on the theoretical aspects of algorithm development and its numerical analysis - Presents a method of different algorithms for specific applications and the relative advantages and limitations of different algorithms for engineers or readers involved in applications - Written by leading subject experts in each field who provide breadth and depth of content coverage
Download or read book Advanced Numerical Approximation of Nonlinear Hyperbolic Equations written by B. Cockburn and published by Springer. This book was released on 2006-11-14 with total page 446 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.
Download or read book Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations written by Xiaobing Feng and published by Springer Science & Business Media. This book was released on 2013-11-08 with total page 289 pages. Available in PDF, EPUB and Kindle. Book excerpt: The field of discontinuous Galerkin finite element methods has attracted considerable recent attention from scholars in the applied sciences and engineering. This volume brings together scholars working in this area, each representing a particular theme or direction of current research. Derived from the 2012 Barrett Lectures at the University of Tennessee, the papers reflect the state of the field today and point toward possibilities for future inquiry. The longer survey lectures, delivered by Franco Brezzi and Chi-Wang Shu, respectively, focus on theoretical aspects of discontinuous Galerkin methods for elliptic and evolution problems. Other papers apply DG methods to cases involving radiative transport equations, error estimates, and time-discrete higher order ALE functions, among other areas. Combining focused case studies with longer sections of expository discussion, this book will be an indispensable reference for researchers and students working with discontinuous Galerkin finite element methods and its applications.
Download or read book Property preserving Numerical Schemes For Conservation Laws written by Dmitri Kuzmin and published by World Scientific. This book was released on 2023-08-28 with total page 491 pages. Available in PDF, EPUB and Kindle. Book excerpt: High-order numerical methods for hyperbolic conservation laws do not guarantee the validity of constraints that physically meaningful approximations are supposed to satisfy. The finite volume and finite element schemes summarized in this book use limiting techniques to enforce discrete maximum principles and entropy inequalities. Spurious oscillations are prevented using artificial viscosity operators and/or essentially nonoscillatory reconstructions.An introduction to classical nonlinear stabilization approaches is given in the simple context of one-dimensional finite volume discretizations. Subsequent chapters of Part I are focused on recent extensions to continuous and discontinuous Galerkin methods. Many of the algorithms presented in these chapters were developed by the authors and their collaborators. Part II gives a deeper insight into the mathematical theory of property-preserving numerical schemes. It begins with a review of the convergence theory for finite volume methods and ends with analysis of algebraic flux correction schemes for finite elements. In addition to providing ready-to-use algorithms, this text explains the design principles behind such algorithms and shows how to put theory into practice. Although the book is based on lecture notes written for an advanced graduate-level course, it is also aimed at senior researchers who develop and analyze numerical methods for hyperbolic problems.
Download or read book Entropy stable Discontinuous Galerkin Finite Element Methods with Streamline Diffusion and Shock capturing for Hyperbolic Systems of Conservation Laws written by Andreas Eduard Hiltebrand and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book SIAM Journal on Scientific Computing written by and published by . This book was released on 2003 with total page 776 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Galerkin Finite Element Methods for Parabolic Problems written by Vidar Thomée and published by Springer Science & Business Media. This book was released on 2010 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Adaptive Discontinuous Galerkin Finite Element Methods for Nonlinear Hyperbolic Conservation Laws written by Ralf Hartmann and published by . This book was released on 2001 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Galerkin Finite Element Methods for Parabolic Problems written by V. Thomee and published by Springer. This book was released on 2006-11-14 with total page 243 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Finite Difference and Discontinuous Galerkin Finite Element Methods for Fully Nonlinear Second Order Partial Differential Equations written by Thomas Lee Lewis and published by . This book was released on 2013 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: The dissertation focuses on numerically approximating viscosity solutions to second order fully nonlinear partial differential equations (PDEs). The primary goals of the dissertation are to develop, analyze, and implement a finite difference (FD) framework, a local discontinuous Galerkin (LDG) framework, and an interior penalty discontinuous Galerkin (IPDG) framework for directly approximating viscosity solutions of fully nonlinear second order elliptic PDE problems with Dirichlet boundary conditions. The developed frameworks are also extended to fully nonlinear second order parabolic PDEs. All of the proposed direct methods are tested using Monge-Ampere problems and Hamilton-Jacobi-Bellman (HJB) problems. Due to the significance of HJB problems in relation to stochastic optimal control, an indirect methodology for approximating HJB problems that takes advantage of the inherent structure of HJB equations is also developed. First, a FD framework is developed that guarantees convergence to viscosity solutions when certain properties concerning admissibility, stability, consistency, and monotonicity are satisfied. The key concepts introduced are numerical operators, numerical moments, and generalized monotonicity. One class of FD methods that fulfills the framework provides a direct realization of the vanishing moment method for approximating second order fully nonlinear PDEs. Next, the emphasis is on extending the FD framework using DG methodologies. In particular, some nonstandard LDG and IPDG methods that utilize key concepts from the FD framework are formulated. Benefits of the DG methodologies over the FD methodology include the ability to handle more complicated domains, more freedom in the design of meshes, higher potential for adaptivity, and the ability to use high order elements as a means for increased accuracy. Last, a class of indirect methods for approximating HJB equations using the vanishing moment method paired with a splitting formulation of the HJB problem is developed and tested numerically. The proposed methodology is well-suited for both continuous and discontinuous Galerkin methods, and it complements the direct methods developed in the dissertation.
Download or read book Partial Differential Equations of Hyperbolic Type and Applications written by Giuseppe Geymonat and published by World Scientific. This book was released on 1987 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the general aspects of hyperbolic conservation laws and their numerical approximation using some of the most modern tools: spectral methods, unstructured meshes and ?-formulation. The applications of these methods are found in some significant examples such as the Euler equations. This book, a collection of articles by the best authors in the field, exposes the reader to the frontier of the research and many open problems.
Download or read book Petroleum Reservoir Simulation written by K. Aziz and published by Springer. This book was released on 1979 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book contains a relatively complete treatment of finite-difference models of black-oil type rservoirs.
Download or read book Adaptive Finite Element Methods for Differential Equations written by Wolfgang Bangerth and published by Springer Science & Business Media. This book was released on 2003-01-23 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. `Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost. At the end of each chapter some exercises are posed in order to assist the interested reader in better understanding the concepts presented. Solutions and accompanying remarks are given in the Appendix.
Download or read book On Cell Entropy Inequality for Discontinuous Galerkin Methods written by Institute for Computer Applications in Science and Engineering and published by . This book was released on 1993 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Galerkin Finite Element Methods for Parabolic Problems written by Vidar Thomee and published by . This book was released on 2014-09-25 with total page 316 pages. Available in PDF, EPUB and Kindle. Book excerpt:
