Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- 1 Introduction -- 1.1 Conservation Laws -- 1.1.1 Integral Form -- 1.1.2 Discontinuous Solutions -- 1.2 Finite Volume Methods -- 1.2.1 Riemann Problems -- 1.2.2 Shock Capturing vs. Tracking -- 1.3 Multidimensional Problems -- 1.4 Linear Waves and Discontinuous Media -- 1.5 CLAWPACK Software -- 1.6 References -- 1.7 Notation -- Part one Linear Equations -- 2 Conservation Laws and Differential Equations -- 2.1 The Advection Equation -- 2.1.1 Variable Coefficients -- 2.2 Diffusion and the Advection-Diffusion Equation -- 2.3 The Heat Equation -- 2.4 Capacity Functions -- 2.5 Source Terms -- 2.5.1 External Heat Sources -- 2.5.2 Reacting Flow -- 2.6 Nonlinear Equations in Fluid Dynamics -- 2.7 Linear Acoustics -- 2.8 Sound Waves -- 2.9 Hyperbolicity of Linear Systems -- 2.9.1 Second-Order Wave Equations -- 2.10 Variable-Coefficient Hyperbolic Systems -- 2.11 Hyperbolicity of Quasilinear and Nonlinear Systems -- 2.12 Solid Mechanics and Elastic Waves -- 2.12.1 Elastic Deformations -- 2.12.2 Strain -- 2.12.3 Stress -- 2.12.4 The Equations of Motion -- 2.13 Lagrangian Gas Dynamics and the Rho-System -- 2.14 Electromagnetic Waves -- Exercises -- 3 Characteristics and Riemann Problems for Linear Hyperbolic Equations -- 3.1 Solution to the Cauchy Problem -- 3.2 Superposition of Waves and Characteristic Variables -- 3.3 Left Eigenvectors -- 3.4 Simple Waves -- 3.5 Acoustics -- 3.6 Domain of Dependence and Range of Influence -- 3.7 Discontinuous Solutions -- 3.8 The Riemann Problem for a Linear System -- 3.9 The Phase Plane for Systems of Two Equations -- 3.9.1 Acoustics -- 3.10 Coupled Acoustics and Advection -- 3.11 Initial-Boundary-Value Problems -- Exercises -- 4 Finite Volume Methods -- 4.1 General Formulation for Conservation Laws.

4.2 A Numerical Flux for the Diffusion Equation -- 4.3 Necessary Components for Convergence -- 4.3.1 Consistency -- 4.4 The CFL Condition -- 4.5 An Unstable Flux -- 4.6 The Lax-Friedrichs Method -- 4.7 The Richtmyer Two-Step Lax-Wendroff Method -- 4.8 Upwind Methods -- 4.9 The Upwind Method for Advection -- 4.10 Godunov's Method for Linear Systems -- Algorithm 4.1 (REA). -- 4.11 The Numerical Flux Function for Godunov's Method -- 4.12 The Wave-Propagation Form of Godunov's Method -- 4.13 Flux-Difference vs. Flux-Vector Splitting -- 4.14 Roe's Method -- Exercises -- 5 Introduction to the CLAWPACK Software -- 5.1 Basic Framework -- 5.2 Obtaining CLAWPACK -- 5.3 Getting Started -- 5.3.1 Creating the Directories -- 5.3.2 Environment variables for the path -- 5.3.3 Compiling the code -- 5.3.4 Makefiles -- 5.3.5 Matlab Graphics -- 5.4 Using CLAWPACK - a Guide through example1 -- 5.4.1 The Main Program (driver.f ) -- 5.4.2 The Initial Conditions (qinit.f ) -- 5.4.3 The claw1ez Routine -- 5.4.4 Boundary Conditions -- 5.4.5 The Riemann Solver -- 5.4.6 The Input File claw1ez.data -- 5.5 Other User-Supplied Routines and Files -- 5.6 Auxiliary Arrays and setaux.f -- 5.7 An Acoustics Example -- Exercises -- 6 High-Resolution Methods -- 6.1 The Lax-Wendroff Method -- 6.2 The Beam-Warming Method -- 6.3 Preview of Limiters -- 6.4 The REA Algorithm with Piecewise Linear Reconstruction -- 6.5 Choice of Slopes -- 6.6 Oscillations -- 6.7 Total Variation -- 6.8 TVD Methods Based on the REA Algorithm -- 6.9 Slope-Limiter Methods -- 6.10 Flux Formulation with Piecewise Linear Reconstruction -- 6.11 Flux Limiters -- 6.12 TVD Limiters -- 6.13 High-Resolution Methods for Systems -- 6.14 Implementation -- 6.15 Extension to Nonlinear Systems -- 6.16 Capacity-Form Differencing -- 6.17 Nonuniform Grids -- 6.17.1 High-Resolution Corrections -- Exercises.

7 Boundary Conditions and Ghost Cells -- 7.1 Periodic Boundary Conditions -- 7.2 Advection -- 7.2.1 Outflow Boundaries -- 7.2.2 Inflow Boundary Conditions -- 7.3 Acoustics -- 7.3.1 Nonreflecting Boundary Conditions -- 7.3.2 Incoming Waves -- 7.3.3 Solid Walls -- 7.3.4 Oscillating Walls -- Exercises -- 8 Convergence, Accuracy, and Stability -- 8.1 Convergence -- 8.1.1 Choice of Norms -- 8.2 One-Step and Local Truncation Errors -- 8.3 Stability Theory -- 8.3.1 Contractive Operators -- 8.3.2 Lax-Richtmyer Stability for Linear Methods -- 8.3.3 2-Norm Stability and von Neumann Analysis -- 8.3.4 1-Norm Stability of the Upwind Method -- 8.3.5 Total-Variation Stability for Nonlinear Methods -- 8.4 Accuracy at Extrema -- 8.5 Order of Accuracy Isn't Everything -- 8.6 Modified Equations -- 8.6.1 The Upwind Method -- 8.6.2 Lax-Wendroff Method -- 8.6.3 Beam-Warming Method -- 8.7 Accuracy Near Discontinuities -- Exercises -- 9 Variable-Coefficient Linear Equations -- 9.1 Advection in a Pipe -- 9.2 Finite Volume Methods -- 9.3 The Color Equation -- 9.3.1 High-Resolution Corrections -- 9.3.2 Discontinuous Velocities -- 9.4 The Conservative Advection Equation -- 9.4.1 Conveyer Belts -- 9.4.2 Traffic Flow -- 9.5 Edge Velocities -- 9.5.1 The Color Equation -- 9.5.2 The Conservative Equation -- 9.6 Variable-Coefficient Acoustics Equations -- 9.7 Constant-Impedance Media -- 9.8 Variable Impedance -- 9.9 Solving the Riemann Problem for Acoustics -- 9.10 Transmission and Reflection Coefficients -- 9.11 Godunov's Method -- 9.12 High-Resolution Methods -- 9.13 Wave Limiters -- 9.14 Homogenization of Rapidly Varying Coefficients -- Exercises -- 10 Other Approaches to High Resolution -- 10.1 Centered-in-Time Fluxes -- 10.2 Higher-Order High-Resolution Methods -- 10.3 Limitations of the Lax-Wendroff (Taylor Series) Approach -- 10.4 Semidiscrete Methods plus Time Stepping.

10.4.1 Evolution Equations for the Cell Averages -- 10.4.2 TVD Time Stepping -- 10.4.3 Reconstruction by Primitive Functions -- 10.4.4 ENO Methods -- 10.5 Staggered Grids and Central Schemes -- Exercises -- Part two Nonlinear Equations -- 11 Nonlinear Scalar Conservation Laws -- 11.1 Traffic Flow -- 11.2 Quasilinear Form and Characteristics -- 11.3 Burgers' Equation -- 11.4 Rarefaction Waves -- 11.5 Compression Waves -- 11.6 Vanishing Viscosity -- 11.7 Equal-Area Rule -- 11.8 Shock Speed -- 11.9 The Rankine-Hugoniot Conditions for Systems -- 11.10 Similarity Solutions and Centered Rarefactions -- 11.11 Weak Solutions -- 11.12 Manipulating Conservation Laws -- 11.14 Entropy Functions -- 11.14.1 The Kružkov Entropies -- 11.15 Long-Time Behavior and N-Wave Decay -- Exercises -- 12 Finite Volume Methods for Nonlinear Scalar Conservation Laws -- 12.1 Godunov's Method -- 12.2 Fluctuations, Waves, and Speeds -- 12.3 Transonic Rarefactions and an Entropy Fix -- 12.4 Numerical Viscosity -- 12.5 The Lax-Friedrichs and Local Lax-Friedrichs Methods -- 12.6 The Engquist-Osher method -- 12.7 E-schemes -- 12.8 High-Resolution TVD Methods -- 12.9 The Importance of Conservation Form -- 12.10 The Lax-Wendroff Theorem -- 12.11 The Entropy Condition -- 12.11.1 Entropy Consistency of Godunov's Method -- 12.12 Nonlinear Stability -- 12.12.1 Convergence Notions -- 12.12.2 Compactness -- 12.12.3 Function Spaces -- 12.12.4 Total-Variation Stability -- Exercises -- 13 Nonlinear Systems of Conservation Laws -- 13.1 The Shallow Water Equations -- 13.2 Dam-Break and Riemann Problems -- 13.3 Characteristic Structure -- 13.4 A Two-Shock Riemann Solution -- 13.5 Weak Waves and the Linearized Problem -- 13.6 Strategy for Solving the Riemann Problem -- 13.7 Shock Waves and Hugoniot Loci -- 13.7.1 The All-Shock Riemann Solution -- 13.7.2 The Entropy Condition.

13.8 Simple Waves and Rarefactions -- 13.8.1 Integral Curves -- 13.8.2 Riemann Invariants -- 13.8.3 Simple Waves -- 13.8.4 Genuine Nonlinearity and Linear Degeneracy -- 13.8.5 Centered Rarefaction Waves -- 13.8.6 The All-Rarefaction Riemann Solution -- 13.9 Solving the Dam-Break Problem -- 13.10 The General Riemann Solver for Shallow Water Equations -- 13.11 Shock Collision Problems -- 13.12 Linear Degeneracy and Contact Discontinuities -- 13.12.1 Shallow Water Equations with a Passive Tracer -- 13.12.2 The Riemann Problem and Contact Discontinuities -- Exercises -- 14 Gas Dynamics and the Euler Equations -- 14.1 Pressure -- 14.2 Energy -- 14.3 The Euler Equations -- 14.4 Polytropic Ideal Gas -- 14.5 Entropy -- 14.5.1 Isentropic Flow -- 14.6 Isothermal Flow -- 14.7 The Euler Equations in Primitive Variables -- 14.8 The Riemann Problem for the Euler Equations -- 14.9 Contact Discontinuities -- 14.10 Riemann Invariants -- 14.11 Solution to the Riemann Problem -- 14.12 The Structure of Rarefaction Waves -- 14.13 Shock Tubes and Riemann Problems -- 14.14 Multifluid Problems -- 14.15 Other Equations of State and Incompressible Flow -- 15 Finite Volume Methods for Nonlinear Systems -- 15.1 Godunov's Method -- 15.2 Convergence of Godunov's Method -- 15.3 Approximate Riemann Solvers -- 15.3.1 Linearized Riemann Solvers -- 15.3.2 Roe Linearization -- 15.3.3 Roe Solver for the Shallow Water Equations -- 15.3.4 Roe Solver for the Euler Equations -- 15.3.5 Sonic Entropy Fixes -- The Harten-Hyman Entropy Fix -- Numerical Viscosity -- Harten's Entropy Fix -- The LLF Entropy Fix -- 15.3.6 Failure of Linearized Solvers -- 15.3.7 The HLL and HLLE Solvers -- 15.4 High-Resolution Methods for Nonlinear Systems -- 15.5 An Alternative Wave-Propagation Implementation of Approximate Riemann Solvers -- 15.6 Second-Order Accuracy -- 15.6.1 Two-Step Lax-Wendroff Methods.

15.7 Flux-Vector Splitting.