Front Tracking for Hyperbolic Conservation Laws

1 Introduction.- 1.1 Notes.- 2 Scalar Conservation Laws.- 2.1 Entropy Conditions.- 2.2 The Riemann Problem.- 2.3 Front Tracking.- 2.4 Existence and Uniqueness.- 2.5 Notes.- 3 A Short Course in Difference Methods.- 3.1 ConservativeMethods.- 3.2 Error Estimates.- 3.3 APriori Error Estimates.- 3.4 Measure-Valued Solutions.- 3.5 Notes.- 4 Multidimensional Scalar Conservation Laws.- 4.1 Dimensional SplittingMethods.- 4.2 Dimensional Splitting and Front Tracking.- 4.3 Convergence Rates.- 4.4 Operator Splitting: Diffusion.- 4.5 Operator Splitting: Source.- 4.6 Notes.- 5 The Riemann Problem for Systems.- 5.1 Hyperbolicity and Some Examples.- 5.2 Rarefaction Waves.- 5.3 The Hugoniot Locus: The Shock Curves.- 5.4 The Entropy Condition.- 5.5 The Solution of the Riemann Problem.- 5.6 Notes.- 6 Existence of Solutions of the Cauchy Problem.- 6.1 Front Tracking for Systems.- 6.2 Convergence.- 6.3 Notes.- 7 Well-Posedness of the Cauchy Problem.- 7.1 Stability.- 7.2 Uniqueness.- 7.3 Notes.- A Total Variation, Compactness, etc..- A.1 Notes.- B The Method of Vanishing Viscosity.- B.1 Notes.- C Answers and Hints.- References.

From the reviews:

"The book under review provides a self-contained, thorough, and modern account of the mathematical theory of hyperbolic conservation laws. ... gives a detailed treatment of the existence, uniqueness, and stability of solutions to a single conservation law in several space dimensions and to systems in one dimension. This book ... is a timely contribution since it summarizes recent and efficient solutions to the question of well-posedness. This book would serve as an excellent reference for a graduate course on nonlinear conservation laws ... ." (M. Laforest, Computer Physics Communications, Vol. 155, 2003)

"The present book is an excellent compromise between theory and practice. Since it contains a lot of theorems, with full proofs, it is a true piece of mathematical analysis. On the other hand, it displays a lot of details and information about numerical approximation for the Cauchy problem. Thus it will be of interest for a wide audience. Students will appreciate the lively and accurate style ... . this text is suitable for graduate courses in PDEs and numerical analysis." (Denis Serre, Mathematical Reviews, 2003 e)