Numerical approximation of hyperbolic systems of conservation laws / Edwige Godlewski, Pierre-Arnaud Raviart.

Godlewski, Edwige [Browse]
Second edition.
New York : Springer, [2021]
xiii, 840 pages : illustrations ; 24 cm.


  • 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.
  • 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.
  • 9781071613429 (hardcover)
  • 1071613421 (hardcover)
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