这本《守恒定律用的数值法》由美国勒维克著，内容是：The overall emphasis is on studying the mathematical tools that are essential in de-veloping analyzing and successfully using numerical methods for nonlinear systems ofconservation laws particularly for problems involving shock waves. A reasonable un-derstanding of the mathematical structure of these equations and their solutions is firstrequired and Part I of these notes deals with this theory. Part II deals more directly withnumerical methods again with the emphasis on general tools that are of broad use. I have stressed the underlying ideas used in various classes of methods rather than present-ing the most sophisticated methods in great detail. My aim was to provide a sufficientbackground that students could then approach the current research literature with thenecessary tools and understanding.

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