Handbook of Numerical Methods for Hyperbolic Problems - 1st Edition - ISBN: 9780444637895, 9780444637956

Handbook of Numerical Methods for Hyperbolic Problems, Volume 17

1st Edition

Basic and Fundamental Issues

Editors: Remi Abgrall Chi-Wang Shu
Series Editors: Qiang Du Roland Glowinski Michael Hintermüller Endre Suli
eBook ISBN: 9780444637956
Hardcover ISBN: 9780444637895
Imprint: North Holland
Published Date: 23rd November 2016
Page Count: 666
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Table of Contents

  • Introduction
    • Acknowledgements
  • Chapter 1: Introduction to the Theory of Hyperbolic Conservation Laws
    • Abstract
    • 1 Introduction
    • 2 Basic Structure of Hyperbolic Conservation Laws
    • 3 Strictly Hyperbolic Systems in One Spatial Dimension
  • Chapter 2: The Riemann Problem: Solvers and Numerical Fluxes
    • Abstract
    • 1 Preliminaries
    • 2 Exact Solution of the Riemann Problem for the Euler Equations
    • 3 The Roe Approximate Riemann Solver
    • 4 The HLL Approximate Riemann Solver
    • 5 The HLLC Approximate Riemann Solver
    • 6 A Numerical Version of the Osher–Solomon Riemann Solver
    • 7 Other Approaches to Constructing Numerical Fluxes
    • 8 Concluding Remarks
    • Acknowledgements
  • Chapter 3: Classical Finite Volume Methods
    • Abstract
    • 1 Some Philosophical Remarks
    • 2 On the Lax–Wendroff Theorem
    • 3 Historical Remarks
    • 4 Weak Solutions and Finite Volume Methods
    • 5 The Cell-Centred Scheme of Jameson, Schmidt and Turkel
    • 6 Cell-Vertex Schemes on Quadrilateral Grids
    • 7 Finite Volume Methods on Unstructured Grids
  • Chapter 4: Sharpening Methods for Finite Volume Schemes
    • Abstract
    • 1 Introduction
    • 2 Sharpening Methods for Linear Equations
    • 3 Coupling With Hyperbolic Nonlinear Equations
  • Chapter 5: ENO and WENO Schemes
    • Abstract
    • 1 Introduction
    • 2 ENO and WENO Approximations
    • 3 ENO and WENO Schemes for Hyperbolic Conservation Laws
    • 4 Selected Topics of Recent Developments
    • Acknowledgements
  • Chapter 6: Stability Properties of the ENO Method
    • Abstract
    • 1 Introduction
    • 2 The ENO Reconstruction Method
    • 3 Application to Conservation Laws
    • 4 ENO Stability Properties
    • 5 Summary
    • Acknowledgements
  • Chapter 7: Stability, Error Estimate and Limiters of Discontinuous Galerkin Methods
    • Abstract
    • 1 Introduction
    • 2 Implementation of DG Methods
    • 3 Stability
    • 4 Error Estimates
    • 5 Limiters for Discontinuous Galerkin Methods
    • 6 Concluding and Remarks
  • Chapter 8: HDG Methods for Hyperbolic Problems
    • Abstract
    • 1 Introduction
    • 2 The Acoustics Wave Equation
    • 3 The Elastic Wave Equations
    • 4 The Electromagnetic Wave Equations
    • 5 Bibliographic Notes
    • Acknowledgements
  • Chapter 9: Spectral Volume and Spectral Difference Methods
    • Abstract
    • 1 Introduction
    • 2 One-Dimensional Formulations
    • 3 Two-Dimensional Formulation on the Simplex
    • 4 Numerical Examples
    • 5 Conclusions
    • Acknowledgements
  • Chapter 10: High-Order Flux Reconstruction Schemes
    • Abstract
    • 1 Introduction
    • 2 FR in 1D
    • 3 FR in Multidimensions
    • 4 Stability and Accuracy of FR Schemes
    • 5 Implementation
    • 6 Applications
    • 7 Summary
  • Chapter 11: Linear Stabilization for First-Order PDEs
    • Abstract
    • 1 Friedrichs’ Systems
    • 2 Weak Formulation and Well-Posedness for Friedrichs’ Systems
    • 3 Residual-Based Stabilization
    • 4 Boundary Penalty for Friedrichs’ Systems
    • 5 Fluctuation-Based Stabilization
  • Chapter 12: Least-Squares Methods for Hyperbolic Problems
    • Abstract
    • 1 Introduction
    • 2 LSFEM for Hyperbolic Problems
    • 3 Conservation Laws
    • 4 Energy Balances
    • 5 Continuous Least-Squares Principles
    • 6 LSFEM in a Hilbert Space Setting
    • 7 Residual Minimization Methods in a Banach Space Setting
    • 8 LSFEMs Based on Adaptively Weighted L2(Ω) Norms
    • 9 Examples
    • 10 A Summary of Conclusions and Recommendations
    • Acknowledgements
  • Chapter 13: Staggered and Colocated Finite Volume Schemes for Lagrangian Hydrodynamics
    • Abstract
    • 1 Historical Background on Lagrangian Computational Fluid Dynamics
    • 2 Lagrangian Hydrodynamics
    • 3 GCL and Related Discrete Operators
    • 4 Discrete Compatible Staggered Lagrangian Hydrodynamics—SGH
    • 5 Discrete Colocated Lagrangian Hydrodynamics—CLH
    • Acknowledgements
  • Chapter 14: High-Order Mass-Conservative Semi-Lagrangian Methods for Transport Problems
    • Abstract
    • 1 Introduction
    • 2 Mass-Conservative SL Schemes
    • 3 Standard Test Sets
    • 4 Nonlinear Vlasov-SL DG and Incompressible Euler System
    • Acknowledgements
  • Chapter 15: Front-Tracking Methods
    • Abstract
    • 1 Introduction
    • 2 FT as a Numerical Algorithm
    • 3 Scientific Uses of FT
    • 4 Conclusions
    • Acknowledgements
  • Chapter 16: Moretti's Shock-Fitting Methods on Structured and Unstructured Meshes
    • Abstract
    • 1 Introduction
    • 2 Shock-Fitting, Upwinding and Modern Shock-Capturing Schemes
    • 3 Boundary Shock-Fitting
    • 4 Floating Shock-Fitting
    • 5 Shock-Fitting for Unstructured Grids
    • 6 Conclusions
  • Chapter 17: Spectral Methods for Hyperbolic Problems
    • Abstract
    • 1 Introduction
    • 2 The Spectral Expansion
    • 3 Spectral Methods
    • 4 Stability and Convergence of Nonlinear Problems
    • 5 Postprocessing Techniques
  • Chapter 18: Entropy Stable Schemes
    • Abstract
    • 1 Entropic Systems of Conservation Laws
    • 2 Discrete Approximations and Entropy Stability
    • 3 Entropy Stable Schemes for Scalar Conservation Laws
    • 4 Semidiscrete Schemes for Systems of Conservation Laws
    • 5 Fully Discrete Schemes for Systems of Conservation Laws
    • 6 Higher-Order Methods
    • 7 Multidimensional Systems of Conservation Laws
    • Acknowledgements
  • Chapter 19: Entropy Stable Summation-by-Parts Formulations for Compressible Computational Fluid Dynamics
    • Abstract
    • 1 Introduction
    • 2 The Compressible NSE
    • 3 SBP Operators
    • 4 Semidiscrete and Fully Discrete Entropy Analysis
    • 5 Entropy Stable Interior Interface Coupling
    • 6 Entropy Stable Solid Wall Boundary Conditions
    • 7 Entropy Stable WENO Formulations
    • 8 Conservation of Entropy in Curvilinear Coordinates
    • 9 Results: Accuracy and Robustness
    • 10 Conclusions
  • Chapter 20: Central Schemes: A Powerful Black-Box Solver for Nonlinear Hyperbolic PDEs
    • Abstract
    • 1 A Very Brief Theoretical Background
    • 2 Finite-Volume Framework
    • 3 First-Order Upwind Schemes
    • 4 First-Order Central Schemes
    • 5 High-Order Finite-Volume Methods
    • 6 Central-Upwind Schemes
    • Acknowledgements
  • Chapter 21: Time Discretization Techniques
    • Abstract
    • 1 Overview
    • 2 Classical Methods
    • 3 Deferred Correction Methods
    • 4 Strong Stability Preserving Methods
    • 5 Other Numerically Optimized Methods
    • 6 IMEX Methods
    • 7 Exponential Time Differencing
    • 8 Multirate Time Stepping
    • 9 Parallel in Time Methods
  • Chapter 22: The Fast Sweeping Method for Stationary Hamilton–Jacobi Equations
    • Abstract
    • 1 Introduction to Hamilton–Jacobi Equation
    • 2 Survey of Numerical Methods for Hamilton–Jacobi Equations
    • 3 The FSM
    • Acknowledgement
  • Chapter 23: Numerical Methods for Hamilton–Jacobi Type Equations
    • Abstract
    • 1 Introduction and Motivations
    • 2 Basics on Viscosity Solutions
    • 3 Evolutive Problems
    • 4 Stationary Problems
    • 5 High-order Approximation Methods
  • Index

Description

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.

Key Features

  • 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

Readership

Researchers, graduate students and engineers working on the design, analysis and applications of numerical algorithms for solving hyperbolic partial differential equations


Details

No. of pages:
666
Copyright:
© North Holland 2016
Published:
Imprint:
North Holland
eBook ISBN:
9780444637956
Hardcover ISBN:
9780444637895

About the Editors

Remi Abgrall Editor

Rémi Abgrall is a professor at Universität Zürich

Affiliations and Expertise

Universitat Zurich, Switzerland

Chi-Wang Shu Editor

Professor Chi-Wang Shu is a professor at Brown University, RI, USA

Affiliations and Expertise

Brown University, RI, USA

About the Series Editors

Qiang Du Series Editor

Roland Glowinski Series Editor

Michael Hintermüller Series Editor

Endre Suli Series Editor