A ROBUST ARBITRARILY HIGH ORDER TRANSPORT METHOD OF THE CHARACTERISTIC TYPE FOR UNSTRUCTURED TETRAHEDRAL GRIDS.
A ROBUST ARBITRARILY HIGH ORDER TRANSPORT METHOD OF THE CHARACTERISTIC TYPE FOR UNSTRUCTURED TETRAHEDRAL GRIDS.

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Publisher:

Published: 2009

Total Pages:

ISBN-13:

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We present a robust arbitrarily high order transport method of the characteristic type for unstructured tetrahedral grids. Previously encountered difficulties have been addressed through the reformulation of the method based on coordinate transformations, evaluation of the moments balance relation as a linear system of equations involving the expansion coefficients of the projected basis, and the asymptotic expansion of the integral kernels in the thin cell limit. The proper choice of basis functions for the high-order spatial expansion of the solution is discussed and its effect on problems involving scattering discussed. Numerical tests are presented to illustrate the beneficial effect of these improvements, and the improved robustness they yield.

Arbitrarily High Order Transport Method of the Characteristic Type for Tetrahedral Grids
Arbitrarily High Order Transport Method of the Characteristic Type for Tetrahedral Grids

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Publisher:

Published: 2001

Total Pages: 18

ISBN-13:

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A formalism is derived for the Arbitrarily High Order Transport (AHOT) method of the Characteristic type (AHOT-C) in three-dimensional geometry for unstructured grids (UG). The resulting equations are implemented in a computer code, AHOT-C-UG, in the C language. The transport solution on the unstructured grid is stored as two inter-linked lists of cell and face flux moments. This arrangement allows the transport sweep to select the order of evaluation dynamically so that the typical recursive ordering of the discrete ordinate's mesh sweep is maintained without the need to store a precomputed order for each ordinate. The dynamic cell sweep order thus reduces the memory demand without excessively increasing execution time. Comparison of AHOT-C-UG's solutions to fine mesh TORT solutions illustrate high accuracy of the new method. In particular, large half a million cell numerical tests illustrate a convergence rate for the error as O(h), where h is a measure of the longest edge in the tetrahedral grid. Execution time on a 700 MHz Intel Pentium III running Linux 2.4.0 is less than 0.2 ms per cell-angle sweep operation. Also the total memory requirement is of the order of 240 bytes per tetrahedral cell, where 64-bit arithmetic precision is employed throughout.

Handbook of Numerical Methods for Hyperbolic Problems
Handbook of Numerical Methods for Hyperbolic Problems

Author: Remi Abgrall

Publisher: Elsevier

Published: 2016-11-17

Total Pages: 668

ISBN-13: 0444637958

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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

Systems of Conservation Laws
Systems of Conservation Laws

Author: Yuxi Zheng

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 324

ISBN-13: 1461201411

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This work should serve as an introductory text for graduate students and researchers working in the important area of partial differential equations with a focus on problems involving conservation laws. The only requisite for the reader is a knowledge of the elementary theory of partial differential equations. Key features of this work include: * broad range of topics, from the classical treatment to recent results, dealing with solutions to 2D compressible Euler equations * good review of basic concepts (1-D Riemann problems) * concrete solutions presented, with many examples, over 100 illustrations, open problems, and numerical schemes * numerous exercises, comprehensive bibliography and index * appeal to a wide audience of applied mathematicians, graduate students, physicists, and engineers Written in a clear, accessible style, the book emphasizes more recent results that will prepare readers to meet modern challenges in the subject, that is, to carry out theoretical, numerical, and asymptotical analysis.

Discontinuous Galerkin Methods
Discontinuous Galerkin Methods

Author: Bernardo Cockburn

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 468

ISBN-13: 3642597211

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A class of finite element methods, the Discontinuous Galerkin Methods (DGM), has been under rapid development recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simula tion, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. In May 24-26, 1999 we organized in Newport (Rhode Island, USA), the first international symposium on DGM with equal emphasis on the theory, numerical implementation, and applications. Eighteen invited speakers, lead ers in the field, and thirty-two contributors presented various aspects and addressed open issues on DGM. In this volume we include forty-nine papers presented in the Symposium as well as a survey paper written by the organiz ers. All papers were peer-reviewed. A summary of these papers is included in the survey paper, which also provides a historical perspective of the evolution of DGM and its relation to other numerical methods. We hope this volume will become a major reference in this topic. It is intended for students and researchers who work in theory and application of numerical solution of convection dominated partial differential equations. The papers were written with the assumption that the reader has some knowledge of classical finite elements and finite volume methods.

Numerical Methods for Elliptic and Parabolic Partial Differential Equations
Numerical Methods for Elliptic and Parabolic Partial Differential Equations

Author: Peter Knabner

Publisher: Springer Science & Business Media

Published: 2003-06-26

Total Pages: 437

ISBN-13: 038795449X

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This text provides an application oriented introduction to the numerical methods for partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises.

DUNE — The Distributed and Unified Numerics Environment
DUNE — The Distributed and Unified Numerics Environment

Author: Oliver Sander

Publisher: Springer Nature

Published: 2020-12-07

Total Pages: 616

ISBN-13: 3030597024

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The Distributed and Unified Numerics Environment (Dune) is a set of open-source C++ libraries for the implementation of finite element and finite volume methods. Over the last 15 years it has become one of the most commonly used libraries for the implementation of new, efficient simulation methods in science and engineering. Describing the main Dune libraries in detail, this book covers access to core features like grids, shape functions, and linear algebra, but also higher-level topics like function space bases and assemblers. It includes extensive information on programmer interfaces, together with a wealth of completed examples that illustrate how these interfaces are used in practice. After having read the book, readers will be prepared to write their own advanced finite element simulators, tapping the power of Dune to do so.

Automated Solution of Differential Equations by the Finite Element Method
Automated Solution of Differential Equations by the Finite Element Method

Author: Anders Logg

Publisher: Springer Science & Business Media

Published: 2012-02-24

Total Pages: 723

ISBN-13: 3642230997

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This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Following are chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.