Numerical Methods for Conservation Laws
Birkhäuser, 11 нояб. 2013 г. - Всего страниц: 214
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. vVithout 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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Computing Discontinuous Solutions
Conservative Methods for Nonlinear Problems
Approximate Riemann Solvers
High Resolution Methods
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2-shock advection equation Buckley-Leverett equation Burgers cell average Chapter characteristic field characteristics compute conservation law consider contact discontinuity convergence convex define density derive differential equation discontinuous solutions discrete eigenvalues eigenvector entropy condition entropy solution entropy-satisfying Euler equations exact solution example EXERCISE flow flux function gas dynamics genuinely nonlinear gives Godunov's method grid Hugoniot locus initial data integral curve integral form isothermal equations jump Lax-Wendroff Lax-Wendroff method linear advection equation linear system matrix nonlinear systems Note numerical methods obtain particle paths phase plane piecewise constant pmax rarefaction wave requires Riemann problem scalar scalar conservation law scalar equations second order accurate shallow water equations shock speed shock wave shown in Figure smooth solutions ſº solution u(r,t space dimensions system of conservation system of equations term theorem total variation TV(U upwind method vanishing viscosity solution variables vector velocity viscosity solution weak solution