Adaptive multiscale schemes for conservation laws.

*(English)*Zbl 1016.76004
Lecture Notes in Computational Science and Engineering. 27. Berlin: Springer. xiv, 184 p. EUR 49.95/net; sFr. 86.00; £35.00; $ 59.95 (2003).

The major focus of this textbook is on adaptive finite volume schemes for conservation laws based on concepts from multi-resolution analysis. First, the basic properties of conservation laws and Godunov-type finite volume schemes are revisited. Then, the considered multiscale setting based on biorthogonal wavelets is introduced and a grid refinement strategy based on the corresponding local multiscale analysis is presented. These key ingredients are then combined to an efficient adaptive finite volume scheme, for which a rigorous error analysis follows. Also, aspects concerning efficient data structures and memory management are briefly discussed. Finally, the adaptive concepts considered are verified by means of several numerical examples, also involving practically relevant flow problems governed by two-dimensional Euler equations. In summary, the text gives a very well founded insight into the current state-of-the-art in the field of adaptive multiscale schemes accompanied by a thorough theoretical analysis. The text is suitable for mathematicians with some background in numerical analysis.

Reviewer: Michael Schäfer (Darmstadt)

##### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76M12 | Finite volume methods applied to problems in fluid mechanics |

65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |