Numerical Methods for Conservation LawsBirkhäuser, 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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Стр. 7
... equation, a scalar conservation law for a single variable u representing the saturation of water in the rock or sand (u = 0 corresponds to pure oil, u = 1 to pure water). Figure 1.3 shows the solution to this 1D problem at three ...
... equation, a scalar conservation law for a single variable u representing the saturation of water in the rock or sand (u = 0 corresponds to pure oil, u = 1 to pure water). Figure 1.3 shows the solution to this 1D problem at three ...
Стр. 15
... conservation of mass equation (2.8) becomes a scalar conservation law for p, pt -- f(p). = 0. (2.9) More typically the equation (2.8) must be solved in conjunction with equations for the conservation of momentum and energy. These ...
... conservation of mass equation (2.8) becomes a scalar conservation law for p, pt -- f(p). = 0. (2.9) More typically the equation (2.8) must be solved in conjunction with equations for the conservation of momentum and energy. These ...
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... conservation law will be fundamental in later analysis. 2.2 Scalar equations Before tackling the complications of coupled systems of equations, we will first study the case of a scalar equation, where m = 1. An example is the conservation ...
... conservation law will be fundamental in later analysis. 2.2 Scalar equations Before tackling the complications of coupled systems of equations, we will first study the case of a scalar equation, where m = 1. An example is the conservation ...
Стр. 18
... Law of heat conduction” (heat diffuses in much the same way as the chemical concentration), which says that the diffusive flux is simply ... Scalar Conservation Laws We begin our study of conservation 18 2 The Derivation of Conservation Laws.
... Law of heat conduction” (heat diffuses in much the same way as the chemical concentration), which says that the diffusive flux is simply ... Scalar Conservation Laws We begin our study of conservation 18 2 The Derivation of Conservation Laws.
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Содержание
1 | |
2 | |
14 | |
Some Scalar Examples | 41 |
Some Nonlinear Systems | 51 |
Linear Hyperbolic Systems | 58 |
Shocks and the Hugoniot Locus | 70 |
Rarefaction Waves and Integral Curves | 81 |
Computing Discontinuous Solutions | 114 |
Conservative Methods for Nonlinear Problems | 122 |
Godunovs Method | 136 |
Approximate Riemann Solvers | 146 |
Nonlinear Stability | 158 |
High Resolution Methods | 173 |
Semidiscrete Methods | 193 |
Multidimensional Problems | 200 |
The Riemann problem for the Euler equations | 89 |
Numerical Methods for Linear Equations | 97 |
Bibliography | 208 |
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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