Cover -- Contents -- Preface -- Acknowledgements -- Notation -- VariabIes -- Operators -- Subscripts and superscripts -- Others -- Chapter 1. Scalar conservation laws -- 1.1 Definitions and basic notions -- 1.2 The Riemann problem -- 1.3 A linear conservation law: the advection equation -- 1.4 A convex conservation law: the Burgers equation -- 1.5 A concave conservation law: the LWR model -- 1.6 A non-convex conservation law: the Buckley-Leverett equation -- 1.7 Extension to multiple dimensions -- Chapter 2. Hyperbolic systems of conservation laws -- 2.1 Definitions -- 2.2 A linear system: the water hammer equations -- 2.3 Two-phase flow in pipes -- 2.4 A 2x2 model for traffic flow -- 2.5 The open channel flow equations with solute transport -- 2.6 The shallow water equations in two dimensions -- Chapter 3. An outline of Godunov-type schemes -- 3.1 The six steps of Godunov-type algorithms -- 3.2 Lagrangian schemes -- 3.3 Multidimensional problems -- 3.4 Stability constraints -- Chapter 4. The Godunov method for scalar laws in one dimension -- 4.1 The linear advection equation -- 4.2 Application to the inviscid Burgers equation -- 4.3 Application to the LWR model -- 4.4 Application to the Buckley-Leverett equation -- Chapter 5. The Godunov method for systems of conservation laws -- 5.1 Application to the water hammer equations -- 5.2 Application to the simplified model for two-phase flow in pipes -- 5.3 Application to a 2x2 traffic flow model -- 5.4 Application to the open channel flow equations -- Chapter 6. Higher-order schemes -- 6.1 Principle of higher-order schemes -- 6.2 The MUSCUPLM schemes -- 6.3 The PPM scheme -- 6.4 The DPM scheme -- 6.5 Boundary conditions for higher-order schemes -- 6.6 Application example -- Chapter 7. Multidimensional schemes -- 7.1 Multidimensional hyperbolic systems of conservation laws -- 7.2 Alternate directions -- 7.3 The finite volume approach -- 7.4 Wave splitting -- 7.5 Computational examples -- 7.6 Higher-order multidimensional schemes -- Chapter 8. Large-time-step algorithms -- 8.1 Front tracking algorithms -- 8.2 Implicit/explicit methods -- 8.3 The time-line reconstruction method -- 8.4 Computational examples -- Chapter 9. Concluding remarks -- Appendix A. Notions in mathematics -- A.1 Linear algebra -- A.2 Accuracy/consistency, stability, convergence -- Appendix B. Riemann solvers -- B.1 Exact Riemann solvers -- B.2 The HLL Riemann solver -- B.3 Roe's Riemann solver -- B.4 Approximate-state solvers -- Appendix C. Sample codes -- C.1 The code Linadv -- C.2 The code 'Burgers' -- C.3 The code 'LWR' -- C.4 The code 'BL' -- C.5 The code 'WatHam' -- C.6 The code '2phase' -- C.7 The code 'Traffic' -- C.8 The code 'Channel' -- C.9 The code 'Sh2D' -- C.10 The code 'Large' -- References -- Index -- L.