守恒定律用的数值法

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

Linear Hyperbolic Systems

6.1 Chaxacteristic 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 Hvgoniot 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 Riemavn problem

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 Courat-Isaacson-Pees method

13.2 Godunov's method

13.3 Linear systems

13.4 The entropy condition

13.5 Scalar conservation laws

14 Approximate Piemann 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

《守恒定律用的数值法》是由世界图书出版公司出版的。 《守恒定律用的数值法》内容简介：These notes developed from a course on the numerical solution of conservation lawsfirst taught at the University of Washington in the fall of 1988 and then at ETH duringthe following spring.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. Ihave 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.【作者简介】作者：（美国）勒维克（Randall J.LeVeque）