Chapter
3. Strictly Hyperbolic Systems in One Spatial Dimension
Chapter 2: The Riemann Problem: Solvers and Numerical Fluxes
1.1. Definitions and Simple Examples
1.2. Hyperbolic Systems and Finite Volume Methods
2. Exact Solution of the Riemann Problem for the Euler Equations
2.1. Equations and Structure of the Solution
2.2. Pressure and Velocity in the Star Region
2.3. The Complete Solution and the 3D Case
2.4. Uses of the Exact Solution of the Riemann Problem
2.5. Approximate Riemann Solvers: Beware
3. The Roe Approximate Riemann Solver
4. The HLL Approximate Riemann Solver
5. The HLLC Approximate Riemann Solver
5.1. Derivation of the HLLC Flux
5.2. Wave Speed Estimates for HLL and HLLC
6. A Numerical Version of the Osher-Solomon Riemann Solver
7. Other Approaches to Constructing Numerical Fluxes
Chapter 3: Classical Finite Volume Methods
1. Some Philosophical Remarks
2. On the Lax-Wendroff Theorem
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
7.1. Cell-Centred Finite Volume Methods
7.2. Vertex-Centred Finite Volume Methods
Chapter 4: Sharpening Methods for Finite Volume Schemes
2. Sharpening Methods for Linear Equations
2.2. Compression Within a BV Setting
2.3. Inequality and Antidiffusion
2.5. PDE Models and Sharpening Methods
2.6. Nature of the Grid/Mesh
2.7. Interface Reconstruction and VOF
3. Coupling With Hyperbolic Nonlinear Equations
3.1. An Example of Discretization for Compressible Flows With Two Components Separated by a Sharp Interface
3.2. Example of Other Evolution Equation Involving Sharp Interfaces
3.3. Cut-Cells and CFL Condition
Chapter 5: ENO and WENO Schemes
2. ENO and WENO Approximations
3. ENO and WENO Schemes for Hyperbolic Conservation Laws
3.1. Finite Volume Schemes
3.2. Finite Difference Schemes
3.3. Remarks on Multidimensional Problems and Systems
4. Selected Topics of Recent Developments
4.2. Steady State Problems
4.3. Time Discretizations for Convection-Diffusion Problems
4.4. Accuracy Enhancement
Chapter 6: Stability Properties of the ENO Method
2. The ENO Reconstruction Method
2.1. Choosing the Stencil Index
3. Application to Conservation Laws
3.1. Finite Volume Methods
3.3. Convergence of High-Order Schemes
4. ENO Stability Properties
4.1. Immediate Properties
4.1.1. Mesh Invariance and Linearity
4.1.2. Discontinuity Across Cell Edges
4.1.3. Uniform kth-Order Accuracy
4.3. Upper Bound on Jumps
4.4. The ENO TV Conjecture
4.5. Mesh-Dependent Properties
4.5.1. Uniform kth-Order Accuracy up to Discontinuities
4.5.2. Monotonicity in Shocked Cells
4.5.3. Essentially Nonoscillatory
4.6.1. R Is Discontinuous
4.6.2. Inefficient Use of Information
4.6.3. Instabilities in Linear Problems
Chapter 7: Stability, Error Estimate and Limiters of Discontinuous Galerkin Methods
2. Implementation of DG Methods
2.1. Semidiscrete Version
3.1. Linear Stability in L2-Norm
4.1. Scalar Equation with Smooth Solution
4.2. Symmetrizable System with Smooth Solution
4.3. Scalar Equation with Discontinuous Initial Solution
4.4. Other Error Estimates
5. Limiters for Discontinuous Galerkin Methods
5.1. Traditional Limiters
5.2. WENO Reconstruction as a Limiter for the RKDG Method
5.3. Hermite WENO Reconstruction
5.4. A Simple WENO-Type Limiter
5.5. A Simple and Compact HWENO Limiter
6. Concluding and Remarks
Chapter 8: HDG Methods for Hyperbolic Problems
2. The Acoustics Wave Equation
2.1. Spatial Discretization
2.2. Temporal Discretization
2.2.3. Adams-Bashforth Methods
3. The Elastic Wave Equations
3.1. Spatial Discretization
3.2. Local Postprocessing
4. The Electromagnetic Wave Equations
4.1. Numerical Discretization
4.2. Local Postprocessing
5.1. Time-Dependent Wave Propagation
5.2. Time-Harmonic Wave Propagation
5.3. Further Reading Material
Chapter 9: Spectral Volume and Spectral Difference Methods
2. One-Dimensional Formulations
2.3. Equivalence of the SV and SD Methods and Their Stability
3. Two-Dimensional Formulation on the Simplex
3.3. Efficiency and Stability
4.1. Double Mach Reflection
4.2. Rayleigh-Taylor Instability Problem With Solution-Based Grid Adaptation
4.3. Aerodynamic Performance of Flapping Wing
Chapter 10: High-Order Flux Reconstruction Schemes
3.2. Tensor Product Elements
4. Stability and Accuracy of FR Schemes
4.2. von Neumann Analysis
5.2. Salient Aspects of an FR Implementation
6.1. Solving the Euler and Navier-Stokes Equations
6.2. Flow Over a Circular Cylinder
6.3. Flow Over an SD7003 Wing
6.4. T106c Low-Pressure Turbine Cascade
Chapter 11: Linear Stabilization for First-Order PDEs
1.1. Basic Ideas and Model Problem
1.2. Example 1: Advection-Reaction Equation
1.3. Example 2: Maxwell's Equations
2. Weak Formulation and Well-Posedness for Friedrichs Systems
2.2. The Boundary Operators
3. Residual-Based Stabilization
3.1. Least-Squares Formulation
3.2. Least-Squares Approximation Using Finite Elements
3.3. Galerkin/Least-Squares
4. Boundary Penalty for Friedrichs Systems
4.2. Boundary Penalty Method
4.3. Galerkin Least-Squares Stabilization with Boundary Penalty
5. Fluctuation-Based Stabilization
5.1. Abstract Theory for Fluctuation-Based Stabilization
5.2. Continuous Interior Penalty
5.3. Two-Scale Stabilization, Local Projection and Subgrid Viscosity
5.3.1. The Two-Scale Decomposition
5.3.2. Local Projection Stabilization
Chapter 12: Least-Squares Methods for Hyperbolic Problems
2. LSFEM for Hyperbolic Problems
4.1. Energy Balances in Hilbert Spaces
4.2. Energy Balances in Banach Spaces
5. Continuous Least-Squares Principles
5.1. Extension to Time-Dependent Conservation Laws
6. LSFEM in a Hilbert Space Setting
6.2. Nonconforming Methods
6.2.1. Discontinuous LSFEM
7. Residual Minimization Methods in a Banach Space Setting
7.1. An L1(Omega) Minimization Method
7.2. Regularized L1(Omega) Minimization Method
8. LSFEMs Based on Adaptively Weighted L2(Omega) Norms
8.1. An Iteratively Reweighted LSFEM
9.1. Approximation of Smooth Solutions
9.2. Approximation of Discontinuous Solutions
10. A Summary of Conclusions and Recommendations
Chapter 13: Staggered and Colocated Finite Volume Schemes for Lagrangian Hydrodynamics
1. Historical Background on Lagrangian Computational Fluid Dynamics
2. Lagrangian Hydrodynamics
2.1. Physical Conservation Laws Written Under Integral Form
2.2. Thermodynamic Closure
2.3. Physical Conservation Laws Written Under Local Form
2.4. Geometrical Conservation Law
3. GCL and Related Discrete Operators
3.1. Grid Notation and Assumptions
3.2. Compatible Discretization of the GCL
3.3. Discrete Divergence and Gradient Operators
3.4.1. Hourglass Filtering
4. Discrete Compatible Staggered Lagrangian Hydrodynamics-SGH
4.1. Notation and Assumptions
4.2. Semidiscrete Compatible Discretization of the GCL
4.3. Semidiscrete Momentum Equation on the Dual Cell ωp
4.4. Semidiscrete Internal Energy Equation on the Primal Cell ωc
4.5. Compatible Discretization of Additional Subcell Forces
4.5.1. Artificial Viscosity Force
4.6.1. Predictor-Corrector Algorithm
4.6.2. Time Step Monitoring
5. Discrete Colocated Lagrangian Hydrodynamics-CLH
5.1. Notation and Assumptions
5.2. Subcell Force-Based Discretization
5.3. Local Entropy Inequality
5.4. Conservation of Total Energy and Momentum
5.6. First-Order Time Discretization
5.7. Second-Order Extension
Chapter 14: High-Order Mass-Conservative Semi-Lagrangian Methods for Transport Problems
2. Mass-Conservative SL Schemes
2.1. SL Finite Difference WENO Scheme
2.1.1. Mass Conservation, Maximum Principle and Positivity-Preserving Numerical Stability
2.1.2. Extension to 2-D Problems
2.1.3. Comparison with a Mass-Conservative Finite Volume SL Scheme
2.2. Mass-Conservative SL DG Scheme
2.2.1. Mass Conservation, Maximum Principle and Positivity-Preserving Stability and Error Estimate
2.2.2. Extension to 2-D Problems
2.2.3. Comparison with SL Finite Difference and Finite Volume Schemes
3.2. 2-D Linear Passive Advection Problems
4. Nonlinear Vlasov-SL DG and Incompressible Euler System
4.1. Vlasov-Poisson Simulations
4.1.2. Two-Stream Instability (Filbet and Sonnendrücker, 2003)
4.2. Guiding Center Model for a Kelvin-Helmholtz Instability
4.3. 2-D Incompressible Euler (Bell et al., 1989)
Chapter 15: Front-Tracking Methods
2. FT as a Numerical Algorithm
2.2. Application Specific (Client) Algorithms, Nonconservative Tracking
2.2.1. Components, the Front Defined Topology and One-Sided Interpolation
2.2.2. Front States and Front Point Propagate
2.2.3. Ghost States and Nonconservative Tracking
2.3. Client Algorithms, Conservative Tracking
2.3.2. Interior State Propagation, Conservative Tracking
2.3.3. Cut Cell Polyhedral Volumes
2.3.4. Directional Tri Propagate
2.3.5. Cut Cell Top, Side Flux and Lagrangian Surface Integrals
2.4. Geometric (FTI) Algorithms
2.4.1. Interface Smoothing
2.4.2. Elimination of Self-Intersections
2.4.3. Robust Parallel Communication of Front Data
3.2. Verification and Validation Examples
3.3. A Complex Physics Example
Chapter 16: Moretti's Shock-Fitting Methods on Structured and Unstructured Meshes
Shock-Fitting, Upwinding and Modern Shock-Capturing Schemes
Floating Shock-Fitting Results
Shock-Fitting for Unstructured Grids
Unstructured Shock-Fitting: Algorithmic Features
Cell Removal Around the Shock Front
Local Remeshing Around the Shock Front
Computation of the Tangent and Normal Unit Vectors
Solution Update Using the Shock-Capturing Code
Enforcement of the Jump Relations
Interpolation of the Phantom Nodes
Unstructured Shock-Fitting: Applications
Transonic (M∞=0.8) Flow Past the NACA 0012 Airfoil
Planar, Transonic Compressible Point-Source Flow
Type IV Shock-Shock Interaction
Hypersonic (M∞ = 24) and Supersonic (M∞ = 4.04) Three-Dimensional Blunt-Body Flows
Un-Steady, Two-Dimensional Flows
Chapter 17: Spectral Methods for Hyperbolic Problems1
2. The Spectral Expansion
2.3. The Duality Between Modes and Nodes
3.3. Interlude on Polynomial Methods and Boundary Conditions
3.3.1. Strongly Imposed Boundary Conditions
3.3.2. Weakly Imposed Boundary Conditions
4. Stability and Convergence of Nonlinear Problems
4.2. Filtering for Stability
4.3. Vanishing Viscosity Techniques
5. Postprocessing Techniques
5.1. Filtering for Accuracy
5.2. Gegenbauer Reconstruction
Chapter 18: Entropy Stable Schemes
1. Entropic Systems of Conservation Laws
1.3. The One-Dimensional Setup
2. Discrete Approximations and Entropy Stability
3. Entropy Stable Schemes for Scalar Conservation Laws
3.3. Numerical Viscosity I
4. Semidiscrete Schemes for Systems of Conservation Laws
4.1. Entropy Variables (Godunov, 1961; Mock, 1980; see also Godunov and Peshkov, 2008)
4.2. Entropy Conservative Fluxes
4.3. How Much Numerical Viscosity
4.4. Scalar Entropy Stability Revisited
4.5. Numerical Viscosity II
4.6. Entropy Conservative Fluxes-Systems of Conservation Laws
5. Fully Discrete Schemes for Systems of Conservation Laws
5.1. Numerical Viscosity III
7. Multidimensional Systems of Conservation Laws
Chapter 19: Entropy Stable Summation-by-Parts Formulations for Compressible Computational Fluid Dynamics
2.2. Continuous Entropy Analysis
3.2. Complementary Grid and Telescopic Flux Form
3.3. Extension to Multiple Dimensions
3.4. Diagonal-Norm SBP Operators
3.5. The Semidiscrete Operators With Boundary and Interface Conditions
4. Semidiscrete and Fully Discrete Entropy Analysis
4.1. Fully Discrete Operators
5. Entropy Stable Interior Interface Coupling
6. Entropy Stable Solid Wall Boundary Conditions
7. Entropy Stable WENO Formulations
7.1. An Entropy Comparison Approach
8. Conservation of Entropy in Curvilinear Coordinates
8.1. Coordinate Transformations and Geometric Conservation Laws
8.2. Curvilinear Conservation and Stability
9. Results: Accuracy and Robustness
9.2. Computation of a Square Cylinder in Supersonic Free Stream
Chapter 20: Central Schemes: A Powerful Black-Box Solver for Nonlinear Hyperbolic PDEs
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
5.1. Second-Order Upwind Schemes
5.2. Second-Order Nessyahu-Tadmor Scheme
6. Central-Upwind Schemes
6.1. Semidiscrete Central-Upwind Schemes
Chapter 21: Time Discretization Techniques
2.3. Multistage Multistep Methods
2.4. Taylor Series Methods
2.5. Multistage Multiderivative Methods
3. Deferred Correction Methods
4. Strong Stability Preserving Methods
4.2. Optimal Explicit Methods
4.3. Optimal Implicit Methods
4.4. Optimal SSP Runge-Kutta Methods for Linear Constant Coefficient Problems
4.5. Optimal Multistep Runge-Kutta Methods
4.6. Strong Stability Properties of Multiderivative Methods
4.7. Widespread Applicability of SSP Methods
5. Other Numerically Optimized Methods
7. Exponential Time Differencing
8. Multirate Time Stepping
9. Parallel in Time Methods
9.1. Concurrency Across the Method
9.2. Concurrency Across the Time Domain
Chapter 22: The Fast Sweeping Method for Stationary Hamilton-Jacobi Equations
1. Introduction to Hamilton-Jacobi Equation
2. Survey of Numerical Methods for Hamilton-Jacobi Equations
3.1. The FSM on a Rectangular Grid
3.2. The FSM for General Convex Hamilton-Jacobi Equations and on Triangular Meshes
3.3. Extension of the FSM
Chapter 23: Numerical Methods for Hamilton-Jacobi Type Equations
1. Introduction and Motivations
1.1. Front Propagation via Level Set Method
1.2. The Infinite Horizon Problem
2. Basics on Viscosity Solutions
3.1. Monotone Schemes in Differenced Form
3.1.1. Upwind Discretization
3.1.1.1. Construction of the Scheme
3.1.2. Central Discretization
3.1.2.1. Construction of the Scheme
3.2.1. Construction of the Scheme
3.3.1. A Numerical Example
4.1. Discretization in Differenced Form
4.3. Convergence and a Priori Error Estimates
5. High-order Approximation Methods
5.2. High-order FD Schemes
5.3. High-order SL Schemes
5.4. Discontinuous Galerkin
Handbook of Numerical Methods for Hyperbolic Problems
：Basic and Fundamental Issues
( Volume 17 )
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.
