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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... variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2. Simple waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . .
... variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2. Simple waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . .
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... variables, such as mass, momentum, and energy in a fluid dynamics problem. More properly, u} is the density function for the jth state variable, with the interpretation that J. w;(x,t) da is the total quantity of this state variable in ...
... variables, such as mass, momentum, and energy in a fluid dynamics problem. More properly, u} is the density function for the jth state variable, with the interpretation that J. w;(x,t) da is the total quantity of this state variable in ...
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... fluid dynamics are the NavierStokes equations, but these include the effects of fluid viscosity and the resulting flux function depends not only on the state variables but also 2 1 Introduction Applications Numerical Methods.
... fluid dynamics are the NavierStokes equations, but these include the effects of fluid viscosity and the resulting flux function depends not only on the state variables but also 2 1 Introduction Applications Numerical Methods.
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... variables (the specific functional relation depends on the gas and is called the “equation of state”). The ... variables are constant. Across two of these waves there are discontinuities in some of the state variables. A shock wave ...
... variables (the specific functional relation depends on the gas and is called the “equation of state”). The ... variables are constant. Across two of these waves there are discontinuities in some of the state variables. A shock wave ...
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... variables are continuous and there is a smooth transition. This wave is called a rarefaction wave since the density of the gas decreases (the gas is rarefied) as this wave passes through. If we put the initial discontinuity at a = 0 ...
... variables are continuous and there is a smooth transition. This wave is called a rarefaction wave since the density of the gas decreases (the gas is rarefied) as this wave passes through. If we put the initial discontinuity at a = 0 ...
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