《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（Randall J.LeVeque）
目录
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
摘要
iscontinuous solutions of the type shown above clearly do not satisfy the PDE in theclassical sense at all points, since the derivatives are not defined at discontinuities. Weneed to define what we mean by a solution to the conservation law in this case. To findthe correct approach we must first understand the derivation of conservation laws fromphysical principles. We wilI see in Chapter 2 that this leads first to an integral form of theconservation law, and that the differential equation is derived from this only by imposingadditional smoothness assumptions on the solution. The crucial fact is that the integralform continues to be valid even for discontinuous solutions. Unfortunately the integral form is more difficult to work with than the differentialequation, especially when it comes to discretization. Since the PDE continues to holdexcept at discontinuities, another approach is to supplement the differential equations byadditional“jump conditions”that must be satisfied across discontinuities. These can bederived by again appealing to the integral form. To avoid the necessity of explicitly imposing these conditions, we will also introducethe weak form of the differential equations. This again involves integrals and allowsdiscontinuous solutions but is easier to work with than the original integral form of theconservation laws. The weak form will be fundamental in the development and analysisof numerical methods. Another mathematical difficulty that we must face is the possible nonuniqueness ofsolutions. Often there is more than one weak solution to the conservation law with thesame initial data. If our conservation law is to model the real world then clearly onlyone of these is physically relevant. The fact that the equations have other, spurious,solutions is a result of the fact that our equations are only a model of reality and somephysical effects have been ignored. In particular, hyperbolic conservation laws do notinclude diffusive or viscous effects. Recall, for example, that the Euler equations resultfrom the Navier-Stokes equations by ignoring fluid viscosity. Although viscous effectsmay be negligible throughout most of the flow, near discontinuities the effect is alwaysstrong. In fact, the full Navier-Stokes equations have smooth solutions for the simpleflows we are considering, and the apparent discontinuities are in reality thin regions withvery steep gradients. What we hope to model with the Euler equations is the limit ofthis smooth solution as the viscosity parameter approaches zero, which will in fact be oneweak solution of the Euler equations.
