Mathematical and computational methods for compressible flow.

*(English)*Zbl 1028.76001
Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press. xiii, 535 p. (2003).

With this book, the authors present most recent numerical techniques for the solution of complicated, technically relevant compressible flow problems. The presentation covers the development of numerical schemes (finite difference, finite volume, and finite element methods) and their mathematical analysis as well as their verification on test problems. In addition, a comprehensive summary of the mathematical theory of compressible flows is given for both inviscid Euler equations and viscous Navier-Stokes system. The theoretical considerations are collected in the first two chapters where equations with heat conduction are derived based on the transport theorem, and necessary thermodynamic relations are introduced. Apart from that, the first chapter contains a list of results from analysis, function spaces and functional analysis. In the second chapter, a summary of theoretical results is given for Euler equations of compressible flows, or, more generally, for hyperbolic partial differential equations and for compressible Navier-Stokes equations (stationary and non-stationary). The topics include: local existence of smooth solutions, weak solutions, entropy condition. Proofs are generally omitted, but references to recent and classical literature are always provided.

Chapter 3 focuses on finite difference and finite volume methods for nonlinear hyperbolic systems and on Euler equations with the main emphasis on the finite volume method applied to the solution of two- and three-dimensional problems on unstructured meshes. Methods for systems in a single space dimension are considered because they are needed for the construction of numerical fluxes in the finite volume method. Special attention is paid to Osher-Solomon scheme and to adaptive methods. The chapter closes with the presentation of several test problems (channel flow, flow around NACA 0012 airfoil, and flow past a cascade of profiles) which are used to compare different mesh types, adaptation strategies, and error and shock indicators that have been introduced in previous sections.

The last chapter 4 is concerned with numerical solution of viscous compressible flow. Among the methods discussed are the streamline diffusion conforming finite element method, combined finite volume-finite element methods, and the discontinuous Galerkin finite element method. Before the finite element techniques are applied to gas dynamical problems, the basic principles and results of the finite element method are introduced for scalar linear elliptic, parabolic, and hyperbolic equations. The same test cases as in chapter 3 are used to assess the quality of different finite element approaches.

Chapter 3 focuses on finite difference and finite volume methods for nonlinear hyperbolic systems and on Euler equations with the main emphasis on the finite volume method applied to the solution of two- and three-dimensional problems on unstructured meshes. Methods for systems in a single space dimension are considered because they are needed for the construction of numerical fluxes in the finite volume method. Special attention is paid to Osher-Solomon scheme and to adaptive methods. The chapter closes with the presentation of several test problems (channel flow, flow around NACA 0012 airfoil, and flow past a cascade of profiles) which are used to compare different mesh types, adaptation strategies, and error and shock indicators that have been introduced in previous sections.

The last chapter 4 is concerned with numerical solution of viscous compressible flow. Among the methods discussed are the streamline diffusion conforming finite element method, combined finite volume-finite element methods, and the discontinuous Galerkin finite element method. Before the finite element techniques are applied to gas dynamical problems, the basic principles and results of the finite element method are introduced for scalar linear elliptic, parabolic, and hyperbolic equations. The same test cases as in chapter 3 are used to assess the quality of different finite element approaches.

Reviewer: Michael Junk (Kaiserslautern)

##### MSC:

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

76Mxx | Basic methods in fluid mechanics |

76Nxx | Compressible fluids and gas dynamics, general |