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Discontinuous Galerkin Method

Analysis and Applications to Compressible Flow

Authors: Vít Dolejší, Miloslav Feistauer

Publisher: Springer International Publishing

Book Series : Springer Series in Computational Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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.

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Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract

The investigation of convection-diffusion problems is a very topical subject in theoretical as well as applied research. On the one hand, these problems play an important role in fluid dynamics, hydrology, heat and mass transfer, environmental protection, water transfer in soils, porous media flow, but also on the other hand, in financial mathematics or image processing. The complexity of these problems prevent from obtaining their exact solution. Therefore, developing a sufficiently robust, accurate and efficient numerical method for computing approximate solutions of (nonlinear) convection-diffusion equations is a challenging problem.

Vít Dolejší, Miloslav Feistauer

Analysis of the Discontinuous Galerkin Method

Frontmatter

Chapter 2. DGM for Elliptic Problems

Abstract

This chapter concerns in basic aspects of the discontinuous Galerkin method (DGM), which will be treated in an example of a simple problem for the Poisson equation with mixed Dirichlet–Neumann boundary conditions. We introduce the discretization of this problem with the aid of several variants of the DGM. Further, we prove the existence of the approximate solution and derive error estimates. Finally, several numerical examples are presented.

Vít Dolejší, Miloslav Feistauer

Chapter 3. Methods Based on a Mixed Formulation

Abstract

In this chapter we introduce two types of the DG discretization that were derived with the aid of a mixed formulation.

Vít Dolejší, Miloslav Feistauer

Chapter 4. DGM for Convection-Diffusion Problems

Abstract

The next Chaps. 4–6 will be devoted to the DGM for the solution of nonstationary, in general nonlinear, convection-diffusion initial-boundary value problems. Some equations treated here can serve as a simplified model of the Navier–Stokes system describing compressible flow, but the subject of convection-diffusion problems is important for a number of areas in science and technology, as is mentioned in the introduction.

Vít Dolejší, Miloslav Feistauer

Chapter 5. Space-Time Discretization by Multistep Methods

Abstract

In practical computations, it is necessary to discretize nonstationary initial-boundary value problems both in space and time.

Vít Dolejší, Miloslav Feistauer

Chapter 6. Space-Time Discontinuous Galerkin Method

In Chap. 5 we introduced and analyzed methods based on the combination of the DGM space discretization with the backward difference formula in time.

Vít Dolejší, Miloslav Feistauer

Chapter 7. Generalization of the DGM

Abstract

The aim of this chapter is to present some advanced aspects and special techniques of the discontinuous Galerkin method. First, we present the hp-discontinuous Galerkin method. Then the DGM over nonstandard nonsimplicial meshes will be treated. Finally, the effect of numerical integration in the DGM will be analyzed in the case of a nonstationary convection-diffusion problem with nonlinear convection.

Vít Dolejší, Miloslav Feistauer

Applications of the Discontinuous Galerkin Method

Frontmatter

Chapter 8. Inviscid Compressible Flow

Abstract

In previous chapters we introduced and analyzed the discontinuous Galerkin method (DGM) for the numerical solution of several scalar equations. However, many practical problems are described by systems of partial differential equations. In the second part of this book, we present the application of the DGM to solving compressible flow problems. The numerical schemes, analyzed for a scalar equation, are extended to a system of equations and numerically verified. We also deal with an efficient solution of resulting systems of algebraic equations.

Vít Dolejší, Miloslav Feistauer

Chapter 9. Viscous Compressible Flow

Abstract

This chapter is devoted to the numerical simulation of viscous compressible flow. The methods treated here represent the generalization of techniques for solving inviscid flow problems contained in Chap. 8. Viscous compressible flow is described by the continuity equation, the Navier–Stokes equations of motion and the energy equation, to which we add closing thermodynamical relations.

Vít Dolejší, Miloslav Feistauer

Chapter 10. Fluid-Structure Interaction

Abstract

Simulating a flow in time dependent domains is a significant part of fluid-structure interaction. It plays an important role in many disciplines. We mention, for example, construction of airplanes (vibrations of wings) or turbines (vibrations of blades), some problems in civil engineering (interaction of wind with constructions of bridges, TV towers or cooling towers of power stations), car industry (vibrations of various elements of a coachwork), but also in medicine (haemodynamics or flow of air in vocal folds). In a number of these examples the moving medium is a gas and the flow is compressible. For low Mach number flows, incompressible models are often used (as e.g., in Sváček et al., J Fluids Struct 23:391–411, 2007, [266]), but in some cases compressibility plays an important role.

Vít Dolejší, Miloslav Feistauer

Backmatter

Title: Discontinuous Galerkin Method
Authors: Vít Dolejší
Miloslav Feistauer
Copyright Year: 2015
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-19267-3
Print ISBN: 978-3-319-19266-6
DOI: https://doi.org/10.1007/978-3-319-19267-3

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