Numerical Methods for Conservation Laws
Springer, 6 дек. 2012 г. - Всего страниц: 220
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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1shock advection equation approximate Riemann solution Burgers canbe cell averages characteristic field characteristics coefficient compute conservation law conservationlaw contact discontinuity convergence convex define density derive differential equations discrete eigenvalues eigenvector entropy condition entropysatisfying Euler equations exact solution example flow flux function fluxlimiter gas dynamics genuinely nonlinear gives Godunov’s method grid Hugoniot locus initial data integral curve integral form inthe Jacobian matrix jump LaxWendroff LeVeque1 Department linear advection equation linear system Lipschitz continuous modified equation monotone methods nonlinear problems nonlinear systems Note numerical methods numerical solution obtain ofthe piecewise constant piecewise linear RankineHugoniot rarefaction wave requires Riemann problem Roe’s satisfied scalar scalar conservation law second order accurate shallow water equations slope smooth solutions solve stability system of conservation thatthe theorem total variation total variation diminishing tothe truncation error TVD methods upwind method variables velocity viscosity solution weak solution XERCISE