Numerical Methods for Conservation Laws (Lectures in Mathematics)

These Notes Were Developed For A Graduate-level Course On The Theory And Numerical Solution Of Nonlinear Hyperbolic Systems Of Conservation Laws. Part I Deals With The Basic Mathematical Theory Of The Equations: The Notion Of Weak Solutions, Entropy Conditions, And A Detailed Description Of The Wave Structure Of Solutions To The Riemann Problem. The Emphasis Is On Tools And Techniques That Are Indispensable In Developing Good Numerical Methods For Discontinuous Solutions. Part Ii Is Devoted To The Development Of High Resolution Shock-capturing Methods, Including The Theory Of Total Variation Diminishing (tvd) Methods And The Use Of Limiter Functions. The Book Is Intended For A Wide Audience, And Will Be Of Use Both To Numerical Analysts And To Computational Researchers In A Variety Of Applications. By Randall J. Leveque. Cover......Page 1 Title Page......Page 4 Copyright......Page 5 Preface......Page 6 Contents......Page 8 I Mathematical Theory......Page 0 1.1 Conservation laws......Page 14 1.2 Applications......Page 15 1.3 Mathematical difficulties......Page 21 1.4 Numerical difficulties......Page 22 1.5 Some references......Page 25 2.1 Integral and differential forms......Page 27 2.2 Scalar equations......Page 29 2.3 Diffusion......Page 30 3.1 The linear advection equation......Page 32 3.1.1 Domain of dependence......Page 33 3.1.2 Nonsmooth data......Page 34 3.2 Burgers' equation......Page 36 3.3 Shock formation......Page 38 3.4 Weak solutions......Page 40 3.5 The Riemann Problem......Page 41 3.6 Shock speed......Page 44 3.7 Manipulating conservation laws......Page 47 3.8 Entropy conditions......Page 49 3.8.1 Entropy functions......Page 50 4.1 Traffic flow......Page 54 4.1.1 Characteristics and "sound speed"......Page 57 4.2 Two phase flow......Page 61 5.1 The Euler equations......Page 64 5.1.1 Ideal gas......Page 66 5.1.2 Entropy......Page 67 5.2 Isentropic flow......Page 68 5.4 The shallow water equations......Page 69 6.1 Characteristic variables......Page 71 6.3 The wave equation......Page 73 6.4 Linearization of nonlinear systems......Page 74 6.4.1 Sound waves......Page 76 6.5 The Riemann Problem......Page 77 6.5.1 The phase plane......Page 80 7.1 The Hugoniot locus......Page 83 7.2 Solution of the Riemann problem......Page 86 7.3 Genuine nonlinearity......Page 88 7.4 The Lax entropy condition......Page 89 7.5 Linear degeneracy......Page 91 7.6 The Riemann problem......Page 92 8.1 Integral curves......Page 94 8.2 Rarefaction waves......Page 95 8.3 General solution of the Riemann problem......Page 99 8.4 Shock collisions......Page 101 9.1 Contact discontinuities......Page 102 9.2 Solution to the Riemann problem......Page 104 II Numerical Methods......Page 108 10 Numerical Methods for Linear Equations......Page 110 10.1 The global error and convergence......Page 115 10.2 Norms......Page 116 10.3 Local truncation error......Page 117 10.4 Stability......Page 119 10.5 The Lax Equivalence Theorem......Page 120 10.6 The CFL condition......Page 123 10.7 Upwind methods......Page 125 11 Computing Discontinuous Solutions......Page 127 11.1 Modified equations......Page 130 11.1.1 First order methods and diffusion......Page 131 11.1.2 Second order methods and dispersion......Page 132 11.2 Accuracy......Page 134 12 Conservative Methods for Nonlinear Problems......Page 135 12.1 Conservative methods......Page 137 12.2 Consistency......Page 139 12.3 Discrete conservation......Page 141 12.4 The Lax-Wendroff Theorem......Page 142 12.5 The entropy condition......Page 146 13 Godunov's Method......Page 149 13.1 The Courant-Isaacson-Reel method......Page 150 13.2 Godunov's method......Page 151 13.3 Linear systems......Page 153 13.4 The entropy condition......Page 155 13.5 Scalar conservation laws......Page 156 14 Approximate Riemann Solvers......Page 159 14.1 General theory......Page 160 14.1.1 The entropy condition......Page 161 14.2 Roe's approximate Riemann solver......Page 162 14.2.1 The numerical flux function for Roe's solver......Page 163 14.2.2 A sonic entropy fix......Page 164 14.2.3 The scalar case......Page 166 14.2.4 A Roe matrix for isothermal flow......Page 169 15.1 Convergence notions......Page 171 15.2 Compactness......Page 172 15.3 Total variation stability......Page 175 15.5 Monotonicity preserving methods......Page 178 15.6 11-contracting numerical methods......Page 179 15.7 Monotone methods......Page 182 16.1 Artificial Viscosity......Page 186 16.2 Flux-limiter methods......Page 189 16.2.1 Linear systems......Page 195 16.3 Slope-limiter methods......Page 196 16.3.1 Linear Systems......Page 200 16.3.2 Nonlinear scalar equations......Page 201 16.3.3 Nonlinear Systems......Page 204 17.1 Evolution equations for the cell averages......Page 206 17.2 Spatial accuracy......Page 208 17.3 Reconstruction by primitive functions......Page 209 17.4 ENO schemes......Page 211 18 Multidimensional Problems......Page 213 18.1 Semi-discrete methods......Page 214 18.2 Splitting methods......Page 215 18.4 Multidimensional approaches......Page 219 Bibliography......Page 221 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. This is a very good book, and covers all the main issues. It is clear and rigorous. Some topics are covered in brief and additional references may be needed to fully understand the topic.