Edwige Godlewski - Numerical Approximation of Hyperbolic Systems of Conservation Laws

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There was an obvious need to complete the first edition of this textbook with the treatment of source terms. Thus, a new chapter (Chap. VII) has been added, which also provides a few important principles concerning nonconservative systems that are naturally introduced with the derivation of well-balanced or asymptotic preserving schemes. Note that most theoretical results are only referred to since it is out of scope to give detailed proofs; these may be tricky and are often quite technical.

We took the opportunity of this second edition to include more examples in the introduction chapter (now Chap. I), such as MHD, shallow water, and flow in a nozzle, and to give some insights on multiphase flow models; this last subject deserves a much longer treatment. Then we thought it is important to emphasize the change of frame from Eulerian to Lagrangian coordinates and the specificity of fluid systems. Additionally, the low Mach limit has been addressed in the chapter devoted to multidimensional systems (now Chap. V) with the final section introducing all Mach schemes.

For 25 years, there has been a tremendous lot of work dedicated to the numerical approximation of hyperbolic systems, among which we choose to introduce the relaxation approach, now at the end of Chap. IV and the case of discontinuous fluxes, and interface coupling, a topic covered in Chap. VII. Both subjects are treated in some specific outlines.

Then, some complements may be found here and there, such as recalling some results of our earlier publication at the beginning of Chap. IV, or more examples of systems of two equations in Chap. II.

We must finally confess that it took us some time to complete the work of this second edition, for different reasons. In fact, most of this work was achieved several years ago, which may explain why only few very recent results are presented, some of them are just mentioned in the notes at the end of each chapter, to give a hint and provide references where the subject is more thoroughly treated.

This work is devoted to the theory and approximation of nonlinear hyperbolic systems of conservation laws in one or two space variables. It follows directly a previous publication on hyperbolic systems of conservation laws by the same authors, and we shall make frequent references to Godlewski and Raviart (1991) (hereafter noted G.R.), though the present volume can be read independently. This earlier publication, apart from a first chapter, especially covered the scalar case. Thus, we shall detail here neither the mathematical theory of multidimensional