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Numerical methods for compressible flow. (English) Zbl 1036.76035
Neustupa, Jiří(ed.) et al., Mathematical fluid mechanics. Recent results and open questions. Basel: Birkhäuser (ISBN 3-7643-6593-5/hbk). Adv. Math. Fluid Mech., 105-142 (2001).
The paper starts by introducing the basic equations for compressible flows together with some basic concepts, like entropy or Mach number. Possible simplifications of the equations are discussed. Then follows an overview on properties of Euler equations, like known results on existence and uniqueness of entropy solutions for Cauchy problem. The presentation of numerical methods for these equations concentrates on finite volume schemes. The construction of numerical fluxes and the choice of boundary conditions for them on the boundary of the domain are discussed in detail. Shock indicators and superconvergence error indicators for controlling an adaptive mesh refinement are presented. The next topic is the numerical solution of viscous compressible flows. A combined finite volume - finite element approach which is based on the splitting of the equations into an inviscid and a viscous part is described in detail. Also for the viscous equations some techniques for controlling a possible anisotropic adaptive mesh refinement are discussed. Finally, a number of numerical examples, including relevant industrial flow problems, are presented.
Altogether, the paper gives an easily understandable short overview on some numerical methods for compressible flows. For the reader which is interested in more details, I would recommend the book written by the author of the paper together with J. Felcman and I. Straskraba [Mathematical and computational methods for compressible flow. Oxford University Press. xiii (2003; Zbl 1028.76001)].
For the entire collection see [Zbl 0971.00052].
76M12 Finite volume methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics, general