Details

Subject(s)

• Gas dynamics [Browse]
• Conservation laws (Mathematics) [Browse]
• Differential equations, Hyperbolic—Numerical solutions [Browse]

Author

• Raviart, Pierre-Arnaud, 1939- [Browse]

Series

• Applied mathematical sciences (Springer-Verlag New York Inc.) ; 118. [More in this series]
• Applied mathematical sciences, 0066-5452 ; 118

Bibliographic references

Includes bibliographical references (pages 749-829) and index.

Contents

• Introduction
• Definitions and Examples
• Fluid Systems in Eulerian and Lagrangian Frames
• Some Averaged Models : Shallow Water, Flow in a Duct, and Two-Phase Flow
• Weak Solutions of Systems of Conservation Laws
• Characteristics in the Scalar One-Dimensional Case
• Weak Solutions : The Rankine-Hugoniot Condition
• Example of Nonuniqueness of Weak Solutions
• Entropy Solution
• A Mathematical Notion of Entropy
• The Vanishing Viscosity Method
• Existence and Uniqueness of the Entropy Solution in the Scalar Case
• Notes
• Nonlinear Hyperbolic Systems in One Space Dimension
• Linear Hyperbolic Systems with Constant Coefficients
• The Nonlinear Case, Definitions and Examples
• Change of Variables, Change of Frame
• The Gas Dynamics Equations
• Ideal MHD
• Simple Waves and Riemann Invariants
• Rarefaction Waves
• Riemann Invariants
• Shock Waves and Contact Discontinuities Characteristic Curves and Entropy Conditions
• Characteristic Curves
• The Lax Entropy Conditions
• Other Entropy Conditions
• Solution of the Riemann Problem
• Examples of Systems of Two Equations
• The Case of a Linear or a Linearly Degenerate System
• The Riemann Problem for the p-System
• The Riemann Problem for the Barotropic Euler System
• Gas Dynamics and Reacting Flows
• Preliminaries
• Properties of the Physical Entropy
• Ideal Gases
• Entropy Satisfying Shock Conditions
• Reacting Flows : The Chapman-Jouguet Theory
• Reacting Flows : The Z.N.D. Model for Detonations
• Finite Volume Schemes for One-Dimensional Systems
• Generalities on Finite Volume Methods for Systems
• Extension of Scalar Schemes to Systems : Some Examples
• L² Stability
• Dissipation and Dispersion
• Godunov's Method
• Godunov's Method for Systems The Gas Dynamics Equations in a Moving Frame
• Godunov's Method in Lagrangian Coordinates
• Godunov's Method in Eulerian Coordinates (Direct Method)
• Godunov's Method in Eulerian Coordinates (Lagrangian Step + Projection)
• Godunov's Method in a Moving Grid
• Godunov-Type Methods
• Approximate Riemann Solvers and Godunov-Type Methods
• Roe's Method and Variants
• The H.L.L. Method
• Osher's Scheme
• Roe-Type Methods for the Gas Dynamics System
• Roe's Method for the Gas Dynamics Equations : (I) The Ideal Gas Case
• Roe's Method for the Gas Dynamics Equations : (II) The "Real Gas" Case
• A Roe-Type Linearization Based on Shock Curve Decomposition
• Another Roe-Type Linearization Associated with a Path
• The Case of the Gas Dynamics System in Lagrangian Coordinates
• Flux Vector Splitting Methods
• General Formulation
• Application to the Gas Dynamics Equations : (I) Steger and Warming's Approach-- Application to the Gas Dynamics Equations : (II) Van Leer's Approach
• Van Leer's Second-Order Method
• Van Leer's Method for Systems
• Solution of the Generalized Riemann Problem
• The G.R.P. for the Gas Dynamics Equations in Lagrangian Coordinates
• Use of the G.R.P. in van Leer's Method
• Kinetic Schemes for the Euler Equations
• The Boltzmann Equation
• The B.G.K. Model
• The Kinetic Scheme
• Some Extensions of the Kinetic Approach
• Relaxation Schemes
• Introduction to Relaxation
• Model Examples
• A Relaxation Scheme for the Euler System
• The Case of Multidimensional Systems
• Generalities on Multidimensional Hyperbolic Systems
• Definitions
• Characteristics
• Simple Plane Waves
• Shock Waves
• The Gas Dynamics Equations in Two Space Dimensions
• Entropy and Entropy Variables
• Invariance of the Euler Equations
• Eigenvalues
• Characteristics's Approach-- Plane Wave Solutions : Self-Similar Solutions
• Multidimensional Finite Difference Schemes
• Direct Approach
• Dimensional Splitting
• Finite-Volume Methods
• Definition of the Finite-Volume Method
• General Results
• Usual Schemes
• Second-Order Finite-Volume Schemes
• Muscl-Type Schemes
• Other Approaches
• An Introduction to All-Mach Schemes for the System of Gas Dynamics
• The Low Mach Limit of the System of Gas Dynamics
• Asymptotic Analysis of the Semi-Discrete Roe Scheme
• An All-Mach Semi-Discrete Roe Scheme
• Asymptotic Analysis of the Semi-Discrete HLL Scheme
• An All-Mach Semi-Discrete HLL Scheme
• An Introduction to Boundary Conditions
• The Initial Boundary Value Problem in the Linear Case
• Scalar Advection Equations
• One-Dimensional Linear Systems : Linearization
• Multidimensional Linear Systems
• The Nonlinear Approach
• Nonlinear Equations
• Nonlinear Systems
• Gas Dynamics Fluid Boundary (Linearized Approach)
• Solid or Rigid Wall Boundary
• Absorbing Boundary Conditions
• Numerical Treatment
• Finite Difference Schemes
• Finite Volume Approach
• Source Terms
• Introduction to Source Terms
• Some General Considerations for Systems with Source Terms
• Simple Examples of Source Terms in the Scalar Case
• Numerical Treatment of Source Terms
• Examples of Systems with Source Terms
• Systems with Geometric Source Terms
• Nonconservative Systems
• Stationary Waves and Resonance
• Case of a Nozzle with Discontinuous Section
• The Example of the Shallow Water System
• Specific Numerical Treatment of Source Terms
• Some Numerical Considerations for Flow in a Nozzle
• Preserving Equilibria, Well-Balanced Schemes
• Schemes for the Shallow Water System
• Simple Approximate Riemann Solvers
• Definition of Simple Approximate Riemann Solvers
• Well-Balanced Simple Schemesynamics Simple Approximate Riemann Solvers in Lagrangian or Eulerian Coordinates
• The Example of the Gas Dynamics Equations with Gravity and Friction
• Link with Relaxation Schemes
• Stiff Source Terms, Asymptotic Preserving Numerical Schemes
• Some Simple Examples
• Derivation of an AP Scheme for the Linear Model
• Euler System with Gravity and Friction
• Interface Coupling
• Introduction to Interface Coupling
• The Interface Coupling Condition
• Numerical Coupling
• References
• Index.

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ISBN

• 9781071613429 (hardcover)
• 1071613421 (hardcover)

OCLC

1255873527

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