Author: Anja Jeschke
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Author: Abdul A. Khan
Publisher: CRC Press
ISBN: 1482226022
Category : Science
Languages : en
Pages : 208

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Book Description
This book introduces the discontinuous Galerkin (DG) method and its application to shallow water flows. The emphasis is to show details and modifications required to apply the scheme to real-world flow problems. It allows the readers to understand and develop robust and efficient computer simulation models that can be used to model flow, contaminant transport, and other factors in rivers and coastal environments. The book includes a large set of tests to illustrate the use of the model for a wide range of applications.

Author: Vít Dolejší
Publisher: Springer
ISBN: 3319192671
Category : Mathematics
Languages : en
Pages : 575

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Book Description
The subject of the book is the mathematical theory of the discontinuous Galerkin method (DGM), which is a relatively new technique for the numerical solution of partial differential equations. The book is concerned with the DGM developed for elliptic and parabolic equations and its applications to the numerical simulation of compressible flow. It deals with the theoretical as well as practical aspects of the DGM and treats the basic concepts and ideas of the DGM, as well as the latest significant findings and achievements in this area. The main benefit for readers and the book’s uniqueness lie in the fact that it is sufficiently detailed, extensive and mathematically precise, while at the same time providing a comprehensible guide through a wide spectrum of discontinuous Galerkin techniques and a survey of the latest efficient, accurate and robust discontinuous Galerkin schemes for the solution of compressible flow.

Author: Bernardo Cockburn
Publisher: Springer Science & Business Media
ISBN: 3642597211
Category : Mathematics
Languages : en
Pages : 468

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Book Description
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.

Author: Dale R. Durran
Publisher: Springer Science & Business Media
ISBN: 1441964126
Category : Mathematics
Languages : en
Pages : 527

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Book Description
This scholarly text provides an introduction to the numerical methods used to model partial differential equations, with focus on atmospheric and oceanic flows. The book covers both the essentials of building a numerical model and the more sophisticated techniques that are now available. Finite difference methods, spectral methods, finite element method, flux-corrected methods and TVC schemes are all discussed. Throughout, the author keeps to a middle ground between the theorem-proof formalism of a mathematical text and the highly empirical approach found in some engineering publications. The book establishes a concrete link between theory and practice using an extensive range of test problems to illustrate the theoretically derived properties of various methods. From the reviews: "...the books unquestionable advantage is the clarity and simplicity in presenting virtually all basic ideas and methods of numerical analysis currently actively used in geophysical fluid dynamics." Physics of Atmosphere and Ocean

Author: Yilong Xiao (Ph. D. in civil engineering)
Publisher:
ISBN:
Category : Civil engineering
Languages : en
Pages : 0

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Book Description
Richards' equation (RE) governs single-phase, variably saturated flows driven by gravity and pressure in porous media and can be used to simulate soil-water flow and associated processes. Solving RE is complicated by factors such as highly nonlinear relationships among soil hydraulic parameters, heterogeneity in soil properties, and the presence of sharp wetting fronts during infiltration into dry soils. These factors lead to two major active challenges: (i) a lack of higher-order accuracy in space, and (ii) a lack of robustness and convergence in implicit time steppers especially under extreme conditions. We tackle these challenges through developing a high-order RE solver based on a local discontinuous Galerkin finite-element method (LDG-FEM or LDG) in space and a dual-time (DT) stepping method in time. LDG achieves high-order accuracy via the use of high-degree basis polynomials, whereas DT resolves lack of convergence in implicit time steppers by turning the original transient problem into a steady-state problem. Our solver is successfully verified against several analytical solutions, showing more resilience and higher accuracy than direct application of the implicit method in the transient form of RE. However, despite the success of DT methods in literature, some of their fundamental aspects such as stability and convergence criteria are inadequately addressed. Often only provided in research articles are recommended ratios of DT step sizes for stability. No conclusion has been reached on how DT convergence rate can be optimized in general, nor is there concrete supportive evidence for the use of high-order pseudo-time schemes given that accuracy is determined by the physical-time scheme. We conducted a stability and convergence analysis to address these aspects for DT methods which couple second-order backward differential formula in physical-time with a fully explicit residual. It is discovered that the ratio of pseudo- to physical-time steps directly influences convergence rate. For linear problems, a closed-formed equation is derived to return the optimal ratio for convergence based on physical parameters.

Author: Stefan Vater
Publisher:
ISBN:
Category : Finite element method
Languages : en
Pages : 40

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Abstract: "In this paper a Godunov-type projection method for computing approximate solutions of the zero Froude number (incompressible) shallow water equations is presented. It is second-order accurate and locally conserves height (mass) and momentum. To enforce the underlying divergence constraint on the velocity field, the predicted numerical fluxes, computed with a standard second order method for hyperbolic conservation laws, are corrected in two steps. First, a MAC-type projection adjusts the advective velocity divergence. In a second projection step, additional momentum flux corrections are computed to obtain new time level cell-centered velocities, which satisfy another discrete version of the divergence constraint. The scheme features an exact and stable second projection. It is obtained by a Petrov-Galerkin finite element ansatz with piecewise bilinear trial functions for the unknown incompressible height and piecewise constant test functions. The stability of the projection is proved using the theory of generalized mixed finite elements, which goes back to Nicolaïdes (1982). In order to do so, the validity of three different inf-sup conditions has to be shown. Since the zero Froude number shallow water equations have the same mathematical structure as the incompressible Euler equations of isentropic gas dynamics, the method can be easily transfered [sic] to the computation of incompressible variable density flow problems."

Author: Clinton N. Dawson
Publisher:
ISBN:
Category : Galerkin methods
Languages : en
Pages :

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Author: Bernardo Cockburn
Publisher:
ISBN:
Category :
Languages : en
Pages : 84

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Author: Jan S. Hesthaven
Publisher: Springer Science & Business Media
ISBN: 0387720650
Category : Mathematics
Languages : en
Pages : 507

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Book Description
This book offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. It covers all key theoretical results, including an overview of relevant results from approximation theory, convergence theory for numerical PDE’s, and orthogonal polynomials. Through embedded Matlab codes, coverage discusses and implements the algorithms for a number of classic systems of PDE’s: Maxwell’s equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations.