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. 8
... obtain an equation with a unique smooth solution, and then let the coefficient of this term go to zero. This “vanishing viscosity” method has some direct uses in the analysis 8 1 Introduction Mathematical difficulties.
... obtain an equation with a unique smooth solution, and then let the coefficient of this term go to zero. This “vanishing viscosity” method has some direct uses in the analysis 8 1 Introduction Mathematical difficulties.
Стр. 9
... obtain numerical results that are very poor. As an example, Figure 1.4 shows some numerical results for the shock tube problem discussed earlier in this chapter. If we use Godunov's method, a first order accurate method described in ...
... obtain numerical results that are very poor. As an example, Figure 1.4 shows some numerical results for the shock tube problem discussed earlier in this chapter. If we use Godunov's method, a first order accurate method described in ...
Стр. 11
... obtained on the shock tube problem with one of the high resolution methods described in Chapter 16. Understanding these methods requires a good understanding of the mathematical theory of conservation laws, as well as some physical ...
... obtained on the shock tube problem with one of the high resolution methods described in Chapter 16. Understanding these methods requires a good understanding of the mathematical theory of conservation laws, as well as some physical ...
Стр. 12
... obtained by solving a Riemann problem with data ul = Us and ur = U#1. Many numerical methods make use of these ... obtain some convergence results and also guarantees that spurious oscillations are not generated. Methods with this ...
... obtained by solving a Riemann problem with data ul = Us and ur = U#1. Many numerical methods make use of these ... obtain some convergence results and also guarantees that spurious oscillations are not generated. Methods with this ...
Стр. 14
... obtained by integrating this in time from ti to to, giving an expression for the mass in [æ1, a2] at time to > t. in terms of the mass at time t1 and the total (integrated) flux at each boundary during this time period: so p(t, t2) da ...
... obtained by integrating this in time from ti to to, giving an expression for the mass in [æ1, a2] at time to > t. in terms of the mass at time t1 and the total (integrated) flux at each boundary during this time period: so p(t, t2) da ...
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