## Numerical Methods for Conservation Laws (Google eBook)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. |

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From a learning standpoint, this book is amazing. Its major shortcoming is that it lacks an index and badly needs one.

### Contents

1 | |

III | 2 |

IV | 8 |

V | 9 |

VI | 12 |

VII | 14 |

VIII | 16 |

IX | 17 |

LXIII | 107 |

LXIV | 110 |

LXV | 112 |

LXVI | 114 |

LXVII | 117 |

LXVIII | 118 |

LXIX | 119 |

LXX | 121 |

X | 19 |

XII | 20 |

XIII | 21 |

XIV | 23 |

XV | 25 |

XVI | 27 |

XVII | 28 |

XVIII | 31 |

XIX | 34 |

XX | 36 |

XXI | 37 |

XXII | 41 |

XXIV | 44 |

XXV | 48 |

XXVI | 51 |

XXVIII | 53 |

XXIX | 54 |

XXX | 55 |

XXXI | 56 |

XXXIII | 58 |

XXXV | 60 |

XXXVII | 61 |

XXXVIII | 63 |

XXXIX | 64 |

XL | 67 |

XLI | 70 |

XLIII | 73 |

XLIV | 75 |

XLVI | 76 |

XLVII | 78 |

XLVIII | 79 |

XLIX | 81 |

LI | 82 |

LII | 86 |

LIII | 88 |

LIV | 89 |

LVI | 91 |

LVII | 95 |

LVIII | 97 |

LIX | 102 |

LX | 103 |

LXI | 104 |

LXII | 106 |

LXXI | 122 |

LXXII | 124 |

LXXIII | 126 |

LXXIV | 128 |

LXXV | 129 |

LXXVI | 133 |

LXXVII | 136 |

LXXVIII | 137 |

LXXIX | 138 |

LXXX | 140 |

LXXXI | 142 |

LXXXII | 143 |

LXXXIII | 146 |

LXXXIV | 147 |

LXXXV | 148 |

LXXXVI | 149 |

LXXXVIII | 150 |

LXXXIX | 151 |

XC | 153 |

XCI | 156 |

XCII | 158 |

XCIII | 159 |

XCIV | 162 |

XCV | 165 |

XCVII | 166 |

XCVIII | 169 |

XCIX | 173 |

CI | 176 |

CII | 182 |

CIII | 183 |

CIV | 187 |

CV | 188 |

CVI | 191 |

CVII | 193 |

CIX | 195 |

CX | 196 |

CXI | 198 |

CXII | 200 |

CXIII | 201 |

CXIV | 202 |

CXV | 206 |

208 | |

### Common terms and phrases

1-shock approximate Riemann solution Buckley-Leverett equation Burgers cell average CFL condition Chapter characteristic field characteristics coefficient compute consider contact discontinuity convergence convex define density derive discrete eigenvalues eigenvector entropy condition Euler equations exact solution example EXERCISE finite difference flow fluid flux function gas dynamics gives Godunov's method Hugoniot locus hyperbolic initial data integral curve integral form ISBN Jacobian matrix jump Lax-Friedrichs method Lax-Wendroff Lax-Wendroff method linear advection equation linear system Lipschitz continuous modified equation monotone methods nonlinear problems nonlinear systems Note numerical flux numerical methods numerical solution obtain order method piecewise constant piecewise linear rarefaction wave requires Riemann problem scalar conservation law second order accurate shallow water equations shock speed slope Slope-limiter smooth solutions solution u(x,t solving stability system of conservation system of equations Theorem total variation total variation diminishing truncation error TVD methods Un+l upwind method variables vector velocity viscosity solution weak solution