These notes developed from a course on the numerical solution of conservation laws first taught at the University of Washington in the fall of 1988 and then at ETH during the 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 of conservation laws, particularly for problems involving shock waves. A reasonable un derstanding of the mathematical structure of these equations and their solutions is first required, and Part I of these notes deals with this theory. Part II deals more directly with numerical 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 sufficient background that students could then approach the current research literature with the necessary tools and understanding. Without the wonders of TeX and LaTeX, these notes would never have been put together. The professional-looking results perhaps obscure the fact that these are indeed lecture notes. Some sections have been reworked several times by now, but others are still preliminary. I can only hope that the errors are. not too blatant. Moreover, the breadth and depth of coverage was limited by the length of these courses, and some parts are rather sketchy.
Table of ContentsI 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 58.- 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.