Numerical methods for conservation laws.

*(English)*Zbl 0723.65067
Lectures in Mathematics, ETH Zürich. Basel etc.: Birkhäuser Verlag. 214 p. sFr 38.00; DM 44.00; $ 24.50 (1990).

These lecture notes originate from a course on numerical solution of conservation laws for graduate and postgraduate students. The emphasis is not on providing full details in the subject but rather on explaining tools that are essential in developing, analyzing, and successfully using numerical methods for nonlinear systems of conservation laws, especially for problems involving shock waves. To this aim it is necessary to understand the mathematical structure of these equations and their solutions. Therefore the book is divided in two equal parts, one treating the mathematical theory and one in which numerical methods are developed and analyzed. Rather than taking an approach from the most general situation to specific cases the book tries to explain facts and techniques on physical examples, e.g. isothermal flow, Euler flow, shock tube problems,... Hence already in the introductory chapter applications such as the above mentioned and many more (MHD, meteorology and weather prediction, astrophysical modeling, flow over a wing, multiphase flow in porous media) are stressed. Mathematical and numerical difficulties are briefly explained in this first chapter.

The theoretical part I really starts with chapter 2 and 3 where scalar conservation laws are derived and simple notions and phenomena such as the flux function, diffusion, domain of dependence, development of shocks, weak solutions, shock speed, entropy functions and entropy conditions are introduced. In chapter 4 two simple scalar examples such as traffic and two phase flow are described. The rest of the mathematical first part is then devoted to systems. It starts with the Euler equations for ideal gas and the simpler cases of isentropic and isothermal flow. The shallow water equation is briefly mentioned. Chapter 6 to 9 are used to develop the solution to the general Riemann problem. Again one starts with the simple linear hyperbolic system. Then the 2*2 system of isothermal flow is used to show how to work in the phase plane with the Hugoniot locus and integral curves. The ideas of genuine nonlinearity, linear degeneracy, and the Lax entropy condition are then introduced to arrive at the solution for the general Riemann problem at least for small initial data. In chapter 9 the Riemann problem for the Euler equations is touched upon only so far as the new phenomena of contact discontinuities is explained.

Part II is devoted to numerical methods. In chapter 10 the classical theory of methods for solving linear constant coefficient differential equations is treated, i.e. the concepts of difference schemes, convergence, local truncation error and stability are briefly introduced. Rather than to prove the Lax equivalence theorem in full detail examples are given and parts of proofs are shown which later extend to the nonlinear problem. The idea of the domain of dependence and as a consequence the CFL condition and the idea of upwinding is explained. In a short chapter the problem of computing discontinuities is shown by means of relating a numerical scheme to a modified equation which is better approximated than the original one. By doing this the author can explain very nicely the typical behavior of numerical solutions due to diffusion and dispersion. It is also seen that the \(L_ 1\)-norm is the natural norm. On a simple example with discontinuities in the solution it is shown what type of convergence one can expect. In chapter 12 the problem of convergence to a weak solution and to the viscosity solution is treated. Hence conservative methods and the numerical entropy condition are introduced. In the following two chapters the Godunov method is explained together with generalizations by using approximate Riemann solvers. Chapter 15 deals with nonlinear stability concepts. The emphasis is on total variation stability which is used to prove a convergence result. Other concepts such as monotonicity preservation, \(l_ 1\)-contractivity and monotonicity of a scheme are treated as well. In one chapter some of the so called ‘high resolution’ schemes are presented. Here the concept of artificial viscosity, flux-limiters and slope-limiters are presented. In chapter 17 and 18 brief outlooks are given to semi-discretization and the numerical treatment of multidimensional problems.

This book is a very good, easy to read introduction to this very active field of numerical methods for conservation laws. It does not have an encyclopedic character but tries to develop on simple examples ideas and concepts. For most proofs one is referred to the original literature.

The theoretical part I really starts with chapter 2 and 3 where scalar conservation laws are derived and simple notions and phenomena such as the flux function, diffusion, domain of dependence, development of shocks, weak solutions, shock speed, entropy functions and entropy conditions are introduced. In chapter 4 two simple scalar examples such as traffic and two phase flow are described. The rest of the mathematical first part is then devoted to systems. It starts with the Euler equations for ideal gas and the simpler cases of isentropic and isothermal flow. The shallow water equation is briefly mentioned. Chapter 6 to 9 are used to develop the solution to the general Riemann problem. Again one starts with the simple linear hyperbolic system. Then the 2*2 system of isothermal flow is used to show how to work in the phase plane with the Hugoniot locus and integral curves. The ideas of genuine nonlinearity, linear degeneracy, and the Lax entropy condition are then introduced to arrive at the solution for the general Riemann problem at least for small initial data. In chapter 9 the Riemann problem for the Euler equations is touched upon only so far as the new phenomena of contact discontinuities is explained.

Part II is devoted to numerical methods. In chapter 10 the classical theory of methods for solving linear constant coefficient differential equations is treated, i.e. the concepts of difference schemes, convergence, local truncation error and stability are briefly introduced. Rather than to prove the Lax equivalence theorem in full detail examples are given and parts of proofs are shown which later extend to the nonlinear problem. The idea of the domain of dependence and as a consequence the CFL condition and the idea of upwinding is explained. In a short chapter the problem of computing discontinuities is shown by means of relating a numerical scheme to a modified equation which is better approximated than the original one. By doing this the author can explain very nicely the typical behavior of numerical solutions due to diffusion and dispersion. It is also seen that the \(L_ 1\)-norm is the natural norm. On a simple example with discontinuities in the solution it is shown what type of convergence one can expect. In chapter 12 the problem of convergence to a weak solution and to the viscosity solution is treated. Hence conservative methods and the numerical entropy condition are introduced. In the following two chapters the Godunov method is explained together with generalizations by using approximate Riemann solvers. Chapter 15 deals with nonlinear stability concepts. The emphasis is on total variation stability which is used to prove a convergence result. Other concepts such as monotonicity preservation, \(l_ 1\)-contractivity and monotonicity of a scheme are treated as well. In one chapter some of the so called ‘high resolution’ schemes are presented. Here the concept of artificial viscosity, flux-limiters and slope-limiters are presented. In chapter 17 and 18 brief outlooks are given to semi-discretization and the numerical treatment of multidimensional problems.

This book is a very good, easy to read introduction to this very active field of numerical methods for conservation laws. It does not have an encyclopedic character but tries to develop on simple examples ideas and concepts. For most proofs one is referred to the original literature.

Reviewer: R.Jeltsch (Zürich)

##### MSC:

65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65H10 | Numerical computation of solutions to systems of equations |

35L67 | Shocks and singularities for hyperbolic equations |

35L65 | Hyperbolic conservation laws |

76W05 | Magnetohydrodynamics and electrohydrodynamics |

86A10 | Meteorology and atmospheric physics |

85A30 | Hydrodynamic and hydromagnetic problems in astronomy and astrophysics |

76B10 | Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing |

76T99 | Multiphase and multicomponent flows |

76S05 | Flows in porous media; filtration; seepage |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

35Q15 | Riemann-Hilbert problems in context of PDEs |

35Q35 | PDEs in connection with fluid mechanics |

76L05 | Shock waves and blast waves in fluid mechanics |