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