Euler and Lagrange conservation laws and numerical methods
In this book, the author studies mathematical and numerical methods for conservation laws appearing mainly in hydrodynamics. The matter of the book is divided into six parts. The first part (Chapter 2) is devoted to the presentation of two important models of conservation laws: a model for road traffic and the gas dynamics system. The basic concepts are presented: Galilean invariance of the system through a suitable change of variables (Euler versus Lagrange), stability, well-posedness and hyperbolicity. In Chapter 3, the simple case of a scalar conservation law is analyzed in 1D. The method of characteristics is recalled and combined to the weak formulation for entropic solutions, leading to uniqueness of the solution. Then numerical simulation of solutions is explained through a general finite volume method. Chapter 4 is devoted to theoretical tools necessary to study the more complex case of systems of conservation laws. The importance of convexity is emphasized on two examples: the Saint Venant system and the gas dynamics system. Then entropy and its associated variables are presented with their relation to hyperbolicity. Self-similar solutions are introduced leading to the classification of elementary simple waves: rarefaction wave, contact discontinuity and shock wave, through Rankine-Hugoniot conditions and then to the complete solution of the Riemann's problem. In Chapter 5, the compressible Euler system of gas dynamics is studied in details. Using the arguments developed in Chapter 4 for the resolution of the Riemann's problem, numerical scemes are introduced. First the (Eulerian) Roe scheme, then the Lagrange+projection scheme, for which entropic and stability results are explained. Finally an intermediate scheme the ALE (Arbitrary Lagrangian Eulerian) scheme is presented in the 1D case. Chapter 6 is devoted to a detailed study of two solvers in 1D for Lagrangian systems. In a first step the solution of a linearized Riemann's problem is provided, then the corresponding numerical discretization is analyzed. In the more advanced last Chapter 7, the previous results on Lagrangian systems are extended to dimension \(N>1\), focusing on delicate mesh strategies for which examples of realistic simulations are given. The book is very clear and pleasant to read and it may also be considered as a useful companion book of the previous one by \textit{F. Dubois} and the same author [Systèmes hyperboliques de lois de conservation. Application à la dynamique des gaz. Palaiseau: Les Éditions de l'École Polytechnique (2005; Zbl 1229.35002)], with the same qualities of accuracy and pedagogical skill. It can be warmly recommended to strongly motivated undergraduate students, and to applied mathematicians working in the field of computational mechanics.
scientific article; zbMATH DE number 5702694