Ⅰ Mathematical Theory 1 Introduction 1.1 Conservation laws 1.2 Applications 1.3 Mathematical difficulties 1.4 Numerical difficulties 1.5 Some references 2 The Derivation of Conservation Laws 2.1 Integral and differential forms 2.2 Scalar equations 2.3 Diffusion 3 Scalar Conservation Laws 3.1 The linear advection equation 3.1.1 Domain of dependence 3.1.2 Nonsmooth data 3.2 Burgers' equation 3.3 Shock formation 3.4 Weak solutions 3.5 The Riemann Problem 3.6 Shock speed 3.7 Manipulating conservation laws 3.8 Entropy conditions 3.8.1 Entropy functions 4 Some Scalar Examples 4.1 Traffic flow 4.1.1 Characteristics and "sound speed" 4.2 Two phase flow 5 Some Nonlinear Systems 5.1 The Euler equations 5.1.1 Ideal gas 5.1.2 Entropy 5.2 Isentropic flow 5.3 Isothermal flow 5.4 The shallow water equations 6 Linear Hyperbolic Systems 6.1 Characteristic variables 6.2 Simple waves 6.3 The wave equation 6.4 Linearization of nonlinear systems 6.4.1 Sound waves 6.5 The Riemann Problem 6.5.1 The phase plane 7 Shocks and the Hugoniot Locus 7.1 The Hugoniot locus 7.2 Solution of the Riemann problem 7.2.1 Riemann problems with no solution 7.3 Genuine nonlinearity 7.4 The Lax entropy condition 7.5 Linear degeneracy 7.6 The Riemann problem 8 Rarefaction Waves and Integral Curves 8.1 Integral curves 8.2 Rarefaction waves 8.3 General solution of the Riemann problem 8.4 Shock collisions 9 The Riemann problem for the Euler equations 9.1 Contact discontinuities 9.2 Solution to the Riemann problem Ⅱ Numerical Methods 10 Numerical Methods for Linear Equations 10.1 The global error and convergence 10.2 Norms 10.3 Local truncation error 10.4 Stability 10.5 The Lax Equivalence ThEorem 10.6 The CFL condition 10.7 Upwind methods 11 Computing Discontinuous Solutions 11.1 Modified equations 11.1.1 First order methods and diffusion 11.1.2 Second order methods and dispersion 11.2 Accuracy 12 Conservative Methods for Nonlinear Problems 12.1 Conservative methods 12.2 Consistency 12.3 Discrete conservation 12.4 The Lax-Wendroff Theorem 12.5 The entropy condition 13 Godunov's Method 13.1 The Courant-Isaacson-Rees method 13.2 Godunov's method 13.3 Linear systems 13.4 The entropy condition 13.5 Scalar conservation laws 14 Approximate Riemann Solvers 14.1 General theory 14.1.1 The entropy condition 14.1.2 Modified conservation laws 14.2 Roe's approximate Riemann solver 14.2.1 The numerical flux function for Roe's solver 14.2.2 A sonic entropy fix 14.2.3 The scalar case 14.2.4 A Roe matrix for isothermal flow 15 Nonlinear Stability 15.1 Convergence notions 15.2 Compactness 15.3 Total variation stability 15.4 Total variation diminishing methods 15.5 Monotonicity preserving methods 15.6 L1 contracting numerical methods 15.7 Monotone methods 16 High Resolution Methods 16.1 Artificial Viscosity 16.2 Flux-limiter methods 16.2.1 Linear systems 16.3 Slope-limiter methods 16.3.1 Linear Systems 16.3.2 Nonlinear scalar equations 16.3.3 Nonlinear Systems 17 Semi-discrete Methods 17.1 Evolution equations for the cell averages 17.2 Spatial accuracy 17.3 Reconstruction by primitive functions 17.4 ENO schemes 18 Multidimensional Problems 18.1 Semi-discrete methods 18.2 Splitting methods 18.3 TVD Methods 18.4 Multidimensional approaches Bibliography