Numerical Methods for Conservation Laws

by Randall J. LeVeque

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.

Genres: Mathematics

232 pages, Paperback

First published January 1, 1990