Handbook of Numerical Methods for Hyperbolic Problems

Basic and Fundamental Issues

1st Edition, Volume 17 - November 17, 2016
Editors: Qiang Du, Remi Abgrall, Chi-Wang Shu, Roland Glowinski, Michael Hintermüller, Endre Süli
Language: English
Hardback ISBN:
9 7 8 - 0 - 4 4 4 - 6 3 7 8 9 - 5
eBook ISBN:
9 7 8 - 0 - 4 4 4 - 6 3 7 9 5 - 6

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 applicati… Read more

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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.

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

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Handbook of Numerical Methods for Hyperbolic Problems

Basic and Fundamental Issues

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Qiang Du

Remi Abgrall

Chi-Wang Shu

Michael Hintermüller