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1992 | Buch | 2. Auflage

Introduction Kapitel lesen Erstes Kapitel lesen

Numerical Methods for Conservation Laws

verfasst von: Randall J. LeVeque

Verlag: Birkhäuser Basel

Buchreihe : Lectures in Mathematics ETH Zürich

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Mathematical Theory

Frontmatter
1. Introduction
Abstract
These notes concern the solution of hyperbolic systems of conservation laws. These are time-dependent systems of partial differential equations (usually nonlinear) with a particularly simple structure.
Randall J. LeVeque
2. The Derivation of Conservation Laws
Abstract
To see how conservation laws arise from physical principles, we will begin by deriving the equation for conservation of mass in a one-dimensional gas dynamics problem, for example flow in a tube where properties of the gas such as density and velocity are assumed to be constant across each cross section of the tube. Let x represent the distance along the tube and let p(x, t) be the density of the gas at point x and time t.
Randall J. LeVeque
3. Scalar Conservation Laws
Abstract
We begin our study of conservation laws by considering the scalar case. Many of the difficulties encountered with systems of equations are already encountered here, and a good understanding of the scalar equation is required before proceeding.
Randall J. LeVeque
4. Some Scalar Examples
Abstract
In this chapter we will look at a couple of examples of scalar conservation laws with some physical meaning, and apply the theory developed in the previous chapter. The first of these examples (traffic flow) should also help develop some physical intuition that is applicable to the more complicated case of gas dynamics, with gas molecules taking the place of cars. This application is discussed in much more detail in Chapter 3 of Whitham[97]. The second example (two phase flow) shows what can happen when f is not convex.
Randall J. LeVeque
5. Some Nonlinear Systems
Abstract
Before developing the theory for systems of conservation laws, it is useful to have some specific examples in mind. In this chapter we will derive some systems of conservation laws.
Randall J. LeVeque
6. Linear Hyperbolic Systems
Abstract
In this chapter we begin the study of systems of conservation laws by reviewing the theory of a constant coefficient linear hyperbolic system. Here we can solve the equations explicitly by transforming to characteristic variables. We will also obtain explicit solutions of the Riemann problem and introduce a “phase space” interpretation that will be very useful in our study of nonlinear systems.
Randall J. LeVeque
7. Shocks and the Hugoniot Locus
Abstract
We now return to the nonlinear system ut + f(u) x = 0, where u(itx,t}) ∈ ℝ m . As before we assume strict hyperbolicity, so that f′{u)} has disctinct real eigenvalues λ1(u) <…< λ m (u) and hence linearly independent eigenvectors. We choose a particular basis for these eigenvectors, {r p {u)}p=1 m , usually chosen to be normalized in some manner, e.g. ∥r(itu})∥ ≡ 1.
Randall J. LeVeque
8. Rarefaction Waves and Integral Curves
Abstract
All of the Riemann solutions considered so far have the following property: the solution is constant along all rays of the form x = ξt. Consequently, the solution is a function of x/t alone, and is said to be a “similarity solution” of the PDE.
Randall J. LeVeque
9. The Riemann problem for the Euler equations
Abstract
I will not attempt to present all of the details for the case of the Euler equations. In principle we proceed as in the examples already presented, but the details are messier. Instead, I will concentrate on discussing one new feature seen here, contact discontinuities, and see how we can take advantage of the linear degeneracy of one field to simplify the solution process for a general Riemann problem. Full details are available in many sources, for example [11], [77], [97].
Randall J. LeVeque

Numerical Methods

Frontmatter
10. Numerical Methods for Linear Equations
Abstract
Before studying numerical methods for nonlinear conservation laws, we review some of the basic theory of numerical methods for the linear advection equation and linear hyperbolic systems. The emphasis will be on concepts that carry over to the nonlinear case.
Randall J. LeVeque
11. Computing Discontinuous Solutions
Abstract
For conservation laws we are naturally interested in the difficulties caused by discontinuities in the solution. In the linear theory presented so far we have assumed smooth solutions, and used this in our discussion of the truncation error and convergence proof.
Randall J. LeVeque
12. Conservative Methods for Nonlinear Problems
Abstract
When we attempt to solve nonlinear conservation laws numerically we run into additional difficulties not seen in the linear equation. Moreover, the nonlinearity makes everything harder to analyze. In spite of this, a great deal of progress has been made in recent years.
Randall J. LeVeque
13. Godunov’s Method
Abstract
Recall that one-sided methods cannot be used for systems of equations with eigenvalues of mixed sign. For a linear system of equations we previously obtained a natural generalization of the upwind method by diagonalizing the system, yielding the method (10.60). For nonlinear systems the matrix of eigenvectors is not constant, and this same approach does not work directly. In this chapter we will study a generalization in which the local characteristic structure, now obtained by solving a Riemann problem rather than by diagonalizing the Jacobian matrix, is used to define a natural upwind method. This method was first proposed for gas dynamics calculations by Godunov[24l.
Randall J. LeVeque
14. Approximate Riemann Solvers
Abstract
Godunov’s method, and higher order variations of the method to be discussed later, require the solution of Riemann problems at every cell boundary in each time step. Although in theory these Riemann problems can be solved, in practice doing so is expensive, and typically requires some iteration for nonlinear equations.
Randall J. LeVeque
15. Nonlinear Stability
Abstract
The Lax-Wendroff Theorem presented in Chapter 12 does not say anything about whether the method converges, only that if a sequence of approximations converges then the limit is a weak solution. To guarantee convergence, we need some form of stability, just as for linear problems. Unfortunately, the Lax Equivalence Theorem no longer holds and we cannot use the same approach (which relies heavily on linearity) to prove convergence. In this chapter we consider one form of nonlinear stability that allows us to prove convergence results for a wide class of practical methods. So far, this approach has been completely successful only for scalar problems. For general systems of equations with arbitrary initial data no numerical method has been proved to be stable or convergent, although convergence results have been obtained in some special cases (e.g. [20], [50], [53]).
Randall J. LeVeque
16. High Resolution Methods
Abstract
In the previous chapter, we observed that monotone methods for scalar conservation laws are TVD and satisfy a discrete entropy condition. Hence they converge in a nonoscillatory manner to the unique entropy solution. However, monotone methods are at most first order accurate, giving poor accuracy in smooth regions of the flow. Moreover, shocks tend to be heavily smeared and poorly resolved on the grid. These effects are due to the large amount of numerical dissipation in monotone methods. Some dissipation is obviously needed to give nonoscillatory shocks and to ensure that we converge to the vanishing viscosity solution, but monotone methods go overboard in this direction.
Randall J. LeVeque
17. Semi-discrete Methods
Abstract
The methods discussed so far have all been fully discrete methods, discretized in both space and time. At times it is useful to consider the discretization process in two stages, first discretizing only in space, leaving the problem continuous in time. This leads to a system of ordinary differential equations in time, called the “semi-discrete equations”. We then discretize in time using any standard numerical method for systems of ordinary differential equations. This approach of reducing a PDE to a system of ODEs, to which we then apply an ODE solver, is often called the method of lines.
Randall J. LeVeque
18. Multidimensional Problems
Abstract
Most practical problems are in two or more space dimensions. So far, we have only considered the one-dimensional (1D) problem. To some extent the 1D methods and theory can be applied to problems in more than one space dimension, and some of these extensions will be briefly described in this chapter. We look at the two-dimensional (2D) case to keep the notation simple, but the same ideas can be used in three dimensions as well.
Randall J. LeVeque
Backmatter
Metadaten
Titel
Numerical Methods for Conservation Laws
verfasst von
Randall J. LeVeque
Copyright-Jahr
1992
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-0348-8629-1
Print ISBN
978-3-7643-2723-1
DOI
https://doi.org/10.1007/978-3-0348-8629-1