Numerical Methods for Conservation Laws / Edition 2

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These notes were developed for a graduate-level course on the theory and numerical solution of nonlinear hyperbolic systems of conservation laws. Part I deals with the basic mathematical theory of the equations: the notion of weak solutions, entropy conditions, and a detailed description of the wave structure of solutions to the Riemann problem. The emphasis is on tools and techniques that are indispensable in developing good numerical methods for discontinuous solutions. Part II is devoted to the development of high resolution shock-capturing methods, including the theory of total variation diminishing (TVD) methods and the use of limiter functions. The book is intended for a wide audience, and will be of use both to numerical analysts and to computational researchers in a variety of applications.

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Editorial Reviews

From the Publisher

"The computing community needs a good text on modern methods for conservation laws, and these notes provide an excellent start on that text. Equally important, LeVeque's perspective and writing style make for wonderful reading and learning. (How often do we find important content and good writing in one book?)" —SIAM Review

"The book by Randall LeVeque is among the first that makes the material in this area accessible to first and second year graduate students in the mathematical sciences. It should be an excellent introduction to this topic for any researcher in the mathematical sciences…This book is based on [the] lecture notes of the author and should serve well as a text for a graduate course…There are many interesting exercises that serve to illuminate and expand on the text, and there are also many well-drawn figures." —Bulletin of the AMS

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Product Details

  • ISBN-13: 9783764327231
  • Publisher: Birkhauser Basel
  • Publication date: 9/17/2009
  • Series: Lectures in Mathematics. ETH Zurich (closed) Series
  • Edition description: 2nd ed. 1992. Corr. 3rd printing 2008
  • Edition number: 2
  • Pages: 220
  • Product dimensions: 0.49 (w) x 9.61 (h) x 6.69 (d)

Table of Contents

I Mathematical Theory.- 1 Introduction.- 1.1 Conservation laws.- 1.2 Applications.- 1.3 Mathematical difficulties.- 1.4 Numerical difficulties.- 1.5 Some references.- 2 The Derivation of Conservation Laws.- 2.1 Integral and differential forms.- 2.2 Scalar equations.- 2.3 Diffusion.- 3 Scalar Conservation Laws.- 3.1 The linear advection equation.- 3.1.1 Domain of dependence.- 3.1.2 Nonsmooth data.- 3.2 Burgers’ equation.- 3.3 Shock formation.- 3.4 Weak solutions.- 3.5 The Riemann Problem.- 3.6 Shock speed.- 3.7 Manipulating conservation laws.- 3.8 Entropy conditions.- 3.8.1 Entropy functions.- 4 Some Scalar Examples.- 4.1 Traffic flow.- 4.1.1 Characteristics and “sound speed”.- 4.2 Two phase flow.- 5 Some Nonlinear Systems.- 5.1 The Euler equations.- 5.1.1 Ideal gas.- 5.1.2 Entropy.- 5.2 Isentropic flow.- 5.3 Isothermal flow.- 5.4 The shallow water equations.- 6 Linear Hyperbolic Systems 58.- 6.1 Characteristic variables.- 6.2 Simple waves.- 6.3 The wave equation.- 6.4 Linearization of nonlinear systems.- 6.4.1 Sound waves.- 6.5 The Riemann Problem.- 6.5.1 The phase plane.- 7 Shocks and the Hugoniot Locus.- 7.1 The Hugoniot locus.- 7.2 Solution of the Riemann problem.- 7.2.1 Riemann problems with no solution.- 7.3 Genuine nonlinearity.- 7.4 The Lax entropy condition.- 7.5 Linear degeneracy.- 7.6 The Riemann problem.- 8 Rarefaction Waves and Integral Curves.- 8.1 Integral curves.- 8.2 Rarefaction waves.- 8.3 General solution of the Riemann problem.- 8.4 Shock collisions.- 9 The Riemann problem for the Euler equations.- 9.1 Contact discontinuities.- 9.2 Solution to the Riemann problem.- II Numerical Methods.- 10 Numerical Methods for Linear Equations.- 10.1 The global error and convergence.- 10.2 Norms.- 10.3 Local truncation error.- 10.4 Stability.- 10.5 The Lax Equivalence Theorem.- 10.6 The CFL condition.- 10.7 Upwind methods.- 11 Computing Discontinuous Solutions.- 11.1 Modified equations.- 11.1.1 First order methods and diffusion.- 11.1.2 Second order methods and dispersion.- 11.2 Accuracy.- 12 Conservative Methods for Nonlinear Problems.- 12.1 Conservative methods.- 12.2 Consistency.- 12.3 Discrete conservation.- 12.4 The Lax-Wendroff Theorem.- 12.5 The entropy condition.- 13 Godunov’s Method.- 13.1 The Courant-Isaacson-Rees method.- 13.2 Godunov’s method.- 13.3 Linear systems.- 13.4 The entropy condition.- 13.5 Scalar conservation laws.- 14 Approximate Riemann Solvers.- 14.1 General theory.- 14.1.1 The entropy condition.- 14.1.2 Modified conservation laws.- 14.2 Roe’s approximate Riemann solver.- 14.2.1 The numerical flux function for Roe’s solver.- 14.2.2 A sonic entropy fix.- 14.2.3 The scalar case.- 14.2.4 A Roe matrix for isothermal flow.- 15 Nonlinear Stability.- 15.1 Convergence notions.- 15.2 Compactness.- 15.3 Total variation stability.- 15.4 Total variation diminishing methods.- 15.5 Monotonicity preserving methods.- 15.6 l1-contracting numerical methods.- 15.7 Monotone methods.- 16 High Resolution Methods.- 16.1 Artificial Viscosity.- 16.2 Flux-limiter methods.- 16.2.1 Linear systems.- 16.3 Slope-limiter methods.- 16.3.1 Linear Systems.- 16.3.2 Nonlinear scalar equations.- 16.3.3 Nonlinear Systems.- 17 Semi-discrete Methods.- 17.1 Evolution equations for the cell averages.- 17.2 Spatial accuracy.- 17.3 Reconstruction by primitive functions.- 17.4 ENO schemes.- 18 Multidimensional Problems.- 18.1 Semi-discrete methods.- 18.2 Splitting methods.- 18.3 TVD Methods.- 18.4 Multidimensional approaches.

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