In the authors’ words ”this book is intended as a first introduction to the theory of difference schemes”. The emphasis is, throughout, on the concepts of convergence, consistency and stability, which are treated in a deeper, clearer and more systematic way than that standard in western introductory texts. The pace is leisurely, the style transparent and the examples and problems abound. On the other hand, since little attention is paid to implementation aspects and there are no numerical experiments at all, I feel that the present book - a belated version of the Russian original (1973; Zbl 0285.65003) - cannot be employed on its own as a general text for students.
While the translation reads rather well, the terminology has not always been adapted to that standard in English. The translator uses ”approximation” where ”consistency” is universally admitted, ”discontinuity decay problem” instead of the familiar ”Riemann problem” etc... Nor is the translation free from the dangers of double transliteration. ”Richtmyer” returns from the Russian as ”Richtmeyer”, ”H. Lewy” (of the CFL condition) as ”G. Levy”... By the way, the reference for the CFL paper is to the Russian 1940 translation and not to the 1928 German original.
The book is divided into fourteen chapters, grouped in five parts. The first part deals with difference equations and includes a detailed discussion of the well-posedness conditions for two-point boundary value problems, a material not easily available in other sources.
The second part presents methods for ordinary differential equations and should only be regarded as a stepping stone for the study of the partial difference schemes considered in the remaining parts of the book. In fact, in spite of the fact that part II contains over one hundred pages, the methods described do not go beyond the classical Runge-Kutta formula and the Adams-Bashforth predictors up to order four; the theory does not even cover the classical Dahlquist’s thesis results; stiffness is not mentioned; and the comments on implementation are badly outdated.
Parts III to V, devoted to difference methods for partial differential equations, are constructed around the themes of stability and convergence. The standard elementary stability topics, such as maximum principles, CFL condition, artificial viscosity, von Neumann condition, principle of frozen coefficients... are treated, along with more advanced issues such as the treatment of non-selfadjoint operators and initial- boundary value problems via the notion of family-spectrum due to the authors. However no reference is made to the important extensions of the Godunov-Ryaben’kij theory due to Gustafsson, Kreiss and Sundström. Another conspicuous absence is that of the crucial ideas of numerical dispersion, phase and group velocity, phase errors etc... which could have been easily included when presenting the von Neumann condition.
To circumvent the linear algebra difficulties in two-dimensional situations, the authors only consider the techniques of splitting, ADI schemes, Chebyshev iteration and the so-called Federenko method (a variant of multigrid). There is a chapter on Galerkin/finite element methods and also brief but excellent introduction to the main ideas in the nonlinear hyperbolic field, including conservation laws, jump- conditions, shock fitting, the importance of schemes in conservation form and the construction of the Godunov scheme for the simple model \(u_ t+uu_ x=0\).