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RICHTMYER, R. D. Editor. Proceedings of the Fourth International Conference on Numerical Methods in Fluid Dynamics. “‘Lecture Notes in Physics”. Vol. 35,457 p. Springer, BerlinHeidelberg-New York, 1975,37 DM. THE CONFERENCE was held on 24-28 June 1974 at the University of Colorado (USA). The collection included 64 papers, of which 27 were read by scientists from the USA, 8 from the USSR, *Zh. vJch.9. Mat. mat. Fiz.. 16,4, 1078-1081, 1976.

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5 from the BRD and France, 4 from England and the remainder by representatives from Italy, Holland, Japan, Australia, Canada, Poland, the Argentine and Israel. The reports considered the numerical solutions of a wide range of hydrodynamic problems, and also some aspects of the construction of numerical algorithms. The reports on hydrodyn~ics may be grouped into sections: flows of inviscid fluids with subsonic, transonic and supersonic velocities; flows of viscous compressible and incompressible fluids; the boundary layer; magnetohydrodynamics; the dynamics of a fluid allowing for various physico-chemical processes. The fmite-difference approach dominates the numerical algorithms of the problems discussed. The concept of an inviscid incompressible fluid was assumed in the papers by Leonard on the numerical simulation of the interaction of vortex filaments and vortex rings, and by Korving on the design of the stationary three-dimensional potential flow around the blades of turbomachines and in others. Several papers were devoted to transonic flow. Thus, Mirin and Burstein presented in their paper a finite-difference method of calculating the inviscid flow about aerofoils. The same problem was solved by Gabutti by the method of asymptotic expansions in the hodograph plane. The method of relaxation for bite-~fference schemes of solution was applied to the calculation of two-d~ension~ and three~~en~on~ transonic flows by Schmidt, Rohlfs and Vanino. They considered linearized problems for the flow around profiles, slender wings and the combination of a wing with an infinite cylindrical body, comparing the calculated results with experiment. Several papers were devoted to supersonic flows of inviscid fluids. A. P. Bazzhin presented a calculation of the conical flow around a triangular plate with a shock wave adjacent only to its vertex. The flow past non-conical wings was considered by Jones. He used a model with a vortex sheet separating off the leading edge of the wing. The position of the vortex and the distribution of the circulation of velocity about a wing of “Concorde” type were calculated. The flow around blunt bodies of revolution was considered by V. V. Rusanov. He applied the finite-difference method to the problem of the supersonic axisymmetric flow past power-shape bodies, developing a body of least wave drag. Rizzi, Klavins and Farmer proposed for the solution of problems of the ~~e-d~e~ion~ supersonic flow past bodies by fmite-difference methods, the use of non-orthogonal curvilinear coordinates, tracking the direction of the flow lines. Subsonic zones for one of the velocity components, which could lead to the breakdown of the working of the algorithm, do not appear in them. The paper by Yoretti was devoted to a review of numerical methods in which shock waves occurring within the flow are isolated and regarded as discontinuities. A considerable proportion of the papers were devoted to the study of flows of a viscous fluid. The model of an incompressible viscous fluid is used to solve problems of the steady-state flow past bodies. the outflow from jets, problems of the flows of a fluid in channels etc. The paper by Lugt and Ohring gave a solution of the interesting non-stationary problem of the rotation of a slender elliptic cylinder in a uniform stream of a viscous fluid. The solution gives a non-zero mean value of the lifting power in accordance with the Magnus effect. The paper by Daly

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gave a numerical method of computing the parameters of the pulsatile flow of blood through constricted arteries for the purpose of studying the formation, growth and separation of clots from the walls of arteries. In a number of papers the application of various numerical methods to the solution of boundary value problems for the Navier-Stokes equations was discussed. Thus, the paper by Roache gave an extension of the numerical methods (LAD and NOS) previously developed by the author, in the paper by Becker the method fmite elements was used to solve problems of the three-dimensional flow of a viscous incompressible fluid under turbulent conditions. Compressible viscous flow was considered in several papers. Thus, the paper by Briley and McDon~d gave an implicit difference scheme for computing threed~en~on~ flows of a viscous gas and an example of the calculation of a stationary subsonic flow in a closed channel. The paper by Widhopf and Victoria considered the numerical solution of the Navier-Stokes equations for the supersonic flow past osc~at~g blunt bodies possessing concavity. The paper by Keller was devoted to the computational problems of the calculation of the tree-tensions boundary layer, in which a difference scheme of second-order accuracy, giving rapid convergence of the iterations and convenient for programming is presented. In the paper by Dwyer and Sanders a difference scheme was given for calculating a three-dimensional boundary layer, taking into account the features of convective and diffusion transfer. A fete-deferent method of second-order accuracy on a non-uniform grid was used by Blottner to analyse a turbulent boundary layer. Peters used a scheme of fourth-order accuracy of the step across the boundary layer, leading to a tridiagonal coefficient matrix, and also a second-order scheme, for calculating the multicomponent boundary layer on a flat plate allowing for chemical reactions, diffusion and thermodiffusion. Papers by Herbert and Fasel were devoted to studies of the stability of the boundary layer. Problems of flow separation were considered in the papers by Abbott, Walker and York, and Williams. The paper by Yu. A. Berezin, V. M. Kovenya and N. N. Yanenko was devoted to the solution of magnetohydrodyn~ic problems. They presented a stable algorithm with an explicit difference scheme for computing the two-dimensional steady supersonic flows around bodies of an ionised gas allowing for viscosity and heat conduction in a magnetic field. The paper by Liu and Chu considered a method of computing the flow of a plasma with a toroidal geometry. A number of papers was devoted to the dynamics of a fluid taking into account various physicochemical processes. These included the paper by V. P. Stulov and V. 1. Mirskii on the supersonic flow past blunt bodies allowing for radiation and equilibrium evaporation from the surface of the body. The paper by B. G. Kuznetsov and Yu. P. Zuikov described the model of the motion of a mixture, constructed allowing for diffusion effects and consisting of a viscous incompressible fluid and particles suspended in it, In connection with the design of lasers the papers by Rivard, Butler and Farmer considered the turbulent motion of a chemically reacting multicomponent gas mixture. In the paper by Thompson and Meng the numerical vortex-in-cell method was applied to the investigation of hydrodynamic problems connected with the formation of vortices at the interface of two media. A complex model of the global atmosphere constructed allowing for conden~tion, convection, diffu~on, radiation etc, was considered in the paper by Somerville. A group of papers was devoted to theoretical questions of the co~tNction of nume~c~ algorithms. Thus, in the paper by Lerat and Peyret the problem of spurious oscillations in the

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numerical solution of the equations of gas dynamics in regions with d&continuities was discussed. The paper by 0. M. Belotserkovskii gave a review of a series of papers executed by the “large particle” method for two-dimensional problems of inviscid and viscous subsonic and supersonic flows, and also problems of a rarefied gas. A. A. Gladkov proposed that the mixed problem for the wave equation be solved by means of the potential of a double layer, the density of the potential being determined numerically. The paper by Kreiss contained a review of the results on a problem for hyperbolic systems of equations in a space of one dimension. The difficulties of the transition to a space of two dimensions were demonstrated by the example of the “shallow water” theory. An energy condition of good condition~ity of the problem was given for a certain explicit difference scheme. The paper by V. I. Paasonen, Yu. I. Shokin and N. N. Yanenko presented the conditions of invariance of finite-difference schemes with respect to groups of transformations permitted by the system of two-dimensional non-stationary differential equations of gas dynamics in Eulerian coordinates. Calculations show that such schemes correspond more completely to the nature of the exact solution. The properties of some difference linear operators, which may be essential for the solution of hydrodynamic problems, and other questions, were investigated. There ISno doubt that the collection reviewed, which contains a large number of interesting papers, is deserving of a more detailed examination than can be given in the restricted space of this review. M. M. Vasil’ev, G. P. Voskresenskii Translated by J. Berry

F. CONST~~NESCU. ~ist~b~ti~nen und ihre Anwendungen in der Physic. 144~. B. G. Teubner, 1974.16.80 DM. THIS book is based on a course of lectures given by the author at the University of Munich. It is intended for physicists and mathematici~s interested in questions of quantum field theory. The book is divided into three parts. In Part 1 the necessary topological apparatus, including the formulations of the fundamental concepts of the theory of topological, metrical, linear, normed and countably-normed spaces is extremely briefly explained. Then continuous linear functionals, conjugate spaces, strong and weak topologies, the union of countably-nosed spaces and linear operators are defined. All the statements encountered are proved. In Part 2 the spaces of basic functions Q S, 8’ and their conjugates are briefly discussed, operations with generalized functions are introduced, in particular, convolutions and the Fourier tr~sfo~ation. The properties of generalized functions depending on a parameter, and the method of regularization, are explained in fair detail. In addition, theorems are proved on the structure of generalized functions from S’ and generalized functions concentrated at a point, and also theorems on Fourier transforms, convolutions and Paley-Wiener-Schwartz transforms. The various properties of the generalized functions ztL, (z*iO) b,z -n, rhare discussed as examples. Two theorems on the boundary values of analytic functions of 4n variables, which are the Fourier-Laplace transforms of functions of the class Sr’, where T’is a cone, are also presented there.

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