By Caterina Calgaro, Jean-François Coulombel, Thierry Goudon

ISBN-10: 3764377410

ISBN-13: 9783764377410

This quantity collects the contributions of a convention held in June 2005, on the laboratoire Paul Painlevé (UMR CNRS 8524) in Lille, France. The assembly used to be meant to study scorching themes and destiny tendencies in fluid dynamics, with the target to foster exchanges of theoretical and numerical viewpoints.The content material of the quantity may be cut up into 3 categories:A first set of contributions is dedicated to the outline of the relationship among various types of fluid dynamics. a huge a part of those papers depends upon the dialogue of the modeling concerns, the identity of the proper dimensionless coefficients, and at the actual interpretation of the models.A moment set of contributions considers the query of the soundness of specific buildings in fluid mechanics equations.The 3rd set of contributions is worried with numerical matters; it's certainly a very important problem to layout numerical schemes which are in a position to trap the advanced beneficial properties generated through fluid flows.

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**Example text**

The inherent small scale sensitiveness of weak solutions for the limit system in (1) or (2) makes their numerical approximation a particularly challenging issue. The core of the diﬃculty indeed stems from the property of shock solutions to be regularization dependent: the artiﬁcial dissipation terms induced by numerical methods tend to corrupt the discrete shocks. Large failures in the celebrated Godunov method in the proper capture of shock solutions to (58) are well exempliﬁed in Chalons [6], approximate Riemann solvers grossly fail as well as illustrated in Berthon [1] and Chalons [6].

Equ. 11 (1986), 1135–1203. [6] F. Bouchut, A. Mangeney-Castelnau, B. P. Vilotte. A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water ﬂows. R. Acad. Sci. Paris, s´erie I, 336(6):531–536, (2003). [7] F. Bouchut, M. Westdickenberg. Gravity driven shallow water models for arbitrary topography. Comm. in Math. , 2(3):359–389, (2004). [8] D. Bresch, B. Desjardins. Numerical approximation of compressible ﬂuid models with density dependent viscosity. In preparation (2005).

To reach a good eﬃciency and minimise the storage (in particular in 3D), we use a multigrid algorithm to solve the problem. It is well known that the relaxation smoothers capture very fast the high frequencies on a given grid. So here we take a set of grid for instance on a square cavity from 4 × 4 cells on the coarsest grid up to 1024 × 1024 cells on the ﬁnest grid. So we use a V-cycle starting on the ﬁnest grid with only two iterations of smoother going down to the coarsest grid and only one when going up with the correction on the low frequencies including the mean value.

### Analysis and Simulation of Fluid Dynamics by Caterina Calgaro, Jean-François Coulombel, Thierry Goudon

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