Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- List of Figures -- Preface -- 1 Introduction -- Part I Basic Theory -- 2 Scalar Conservation Laws -- 2.1 Theoretical Background -- Weak Solutions and Jump Conditions -- Shocks, Rarefaction Waves, and Entropy -- The Riemann Problem -- 2.2 Basic Concepts of Numerical Approximation -- Convergence -- 3 The GRP Method for Scalar Conservation Laws -- 3.1 From Godunov to the GRP Method -- 3.2 1-D Sample Problems -- 3.2.1 The Linear Conservation Law -- First-Order Schemes -- Second-Order Schemes -- 3.2.2 The Burgers Nonlinear Conservation Law -- First-Order Computation -- Second-Order Computation -- 3.3 2-D Sample Problems -- The Operator-Splitting Method -- The Linear Conservation Law -- The Nonlinear Burgers Equation -- Case A -- Case B -- Case C -- Case D -- The Guckenheimer Equation -- 4 Systems of Conservation Laws -- 4.1 Nonlinear Hyperbolic Systems in One Space Dimension -- Characteristic Curves and Centered Rarefaction Waves -- Weak Solutions and Jump Discontinuities -- Entropy Conditions, Shock Waves, and Contact Discontinuities -- The Riemann Problem -- 4.2 Euler Equations of Quasi-1-D, Compressible, Inviscid Flow -- The Flow Equations -- Eigenvalues and Characteristic Equations -- Isentropic Flow -- Weak Solutions and Jump Conditions -- Lagrangian Coordinates -- Shock Waves - Detailed Study of the Jump Condition -- Centered Rarefaction Waves -- The Riemann Problem (RP) for Planar Flows -- Perfect (Gamma-Law) Gas -- 5 The Generalized Riemann Problem (GRP) for Compressible Fluid Dynamics -- 5.1 The GRP for Quasi-1-D, Compressible, Inviscid Flow -- Structure of the Solution to the GRP -- The Linear GRP in Lagrangian Coordinates - Setup and Statement of the Main Theorem -- The Acoustic Case -- Resolution of a CRW in the Lagrangian Framework.

Explicit Formulas for the (Lagrangian) GRP in the Gamma-Law Case -- Concluding the Treatment of the CRW -- Time Derivatives of p, u on the Interface - Proof of the Main Theorem -- Conclusion of the Linear GRP in the Lagrangian Case -- The Linear GRP in the Eulerian Framework -- 5.2 The GRP Numerical Method for Quasi-1-D, Compressible, Inviscid Flow -- The Godunov Scheme -- The Basic GRP Scheme -- The E and L Schemes, Intermediate Schemes, and MUSCL -- Updating the Slopes -- Concluding the GRP Algorithm -- 6 Analytical and Numerical Treatment of Fluid Dynamical Problems -- 6.1 The Shock Tube Problem -- 6.2 Wave Interactions -- 6.2.1 Shock-Contact Interaction -- 6.2.2 Shock-Shock Interaction -- 6.2.3 Shock-CRW Interaction -- 6.2.4 CRW-Contact Interaction -- Approximate Analysis of the Interaction -- Numerical (GRP) Solution -- 6.3 Spherically Converging Flow of Cold Gas -- 6.4 The Flow Induced by an Expanding Sphere -- 6.5 Converging-Diverging Nozzle Flow -- Nozzle Geometry and Steady Flow -- The Finite-Difference Solution -- Part II Numerical Implementation -- 7 From the GRP Algorithm to Scientific Computing -- 7.1 General Discussion -- 7.2 Strang's Operator-Splitting Method -- 7.3 Two-Dimensional Flow in Cartesian Coordinates -- The Linear GRP for a Planar System with Advection -- The Split Scheme for (7.16) -- The Split Scheme and Conservation Form -- 8 Geometric Extensions -- 8.1 Grids That Move in Time -- 8.2 Singularity Tracking -- 8.3 Moving Boundary Tracking (MBT) -- 8.3.1 Basic Setup -- 8.3.2 Survey of the Full MBT Algorithm -- 8.3.3 An Example: Shock Lifting of an Elliptic Disk -- 9 A Physical Extension: Reacting Flow -- 9.1 The Equations of Compressible Reacting Flow -- The Characteristic Relations -- Discontinuities and Centered Rarefaction Waves -- 9.2 The Chapman-Jouguet (C-J) Model -- 9.3 The Z-N-D (Zeldovich-von Neumann-Döring) Solution.

9.4 The Linear GRP for the Reacting-Flow System -- The Associated Riemann Problem -- Structure of the Solution to the Linear GRP-The Main Theorem -- The Acoustic Approximation -- Resolution of the Centered Rarefaction Wave -- Conclusion of the Linear GRP -- The Gamma-Law Case -- 9.5 The GRP Scheme for Reacting Flow -- 10 Wave Interaction in a Duct - A Comparative Study -- Appendix A Entropy Conditions for Scalar Conservation Laws -- Appendix B Convergence of the Godunov Scheme -- Appendix C Riemann Solver for a Gamma-Law Gas -- Appendix D The MUSCL Scheme -- Bibliography -- Glossary -- Flow Variables and Thermodynamic Quantities (Section 4.2) -- Coordinates -- Riemann and Generalized Riemann Problem -- Functional Spaces -- Index.