I am very glad that this book is now accessible to English-speaking scientists. During the three years following the publication of the original Russian edition, the method of differential approximation has been rapidly expanded and unfortunately I was unable to incorporate into the English edition all of the material which whould have reflected its present state of development. Nevertheless, a considerable amount of recently obtained results have been added and the bibliography has been enlarged accordingly, so that the English edition is one third longer than the Russian original. Mathematical rigorousness is a basic feature of this monograph. The reader should therefore be familiar with the theory of partial differential equations and difference equations. Some knowledge of group theory as applied to problems in physics, especially the theory of Lie groups, would also be useful. The treatment of the approximation of gas dynamic equations focuses on the question of how to characterize the typical features of difference equations on the basis of the related differential approximation, which can be discussed using the fully developed theory of partial differential equations.

Inhalt
I. Stability Analysis of Difference Schemes by the Method of Differential Approximation.- 1. Certain Properties of the Theory of Linear Differential Equations and Difference Schemes.- 2. The Concept of the Differential Approximation of a Difference Scheme.- 3. Stability Analysis of Difference Schemes with Constant Coefficients by Means of the Differential Representation.- 4. Connection Between The Stability of Difference Schemes and the Properties of Their First Differential Approximations.- 5. Dissipative Difference Schemes for Hyperbolic Equations.- 6. A Means for the Construction of Difference Schemes with Higher Order of Approximation.- II. Investigation of the Artificial Viscosity of Difference Schemes.- 7. K-property of Difference Schemes.- 8. Investigation of Dissipation and Dispersion of Difference Schemes.- 9. Application of the Method of Differential Approximation to the Investigation of the Effects of Nonlinear Transformations.- 10. Investigation of Monotonicity of Difference Schemes.- 11. Difference Schemes in an Arbitrary Curvilinear Coordinate System.- III. Invariant Difference Schemes.- 12. Some Basic Concepts of the Theory of Group Properties of Differential Equations.- 13. Groups Admitted by the System of the Equations of Gas Dynamics.- 14. A Necessary and Sufficient Condition for Invariance of Difference Schemes on the Basis of the First Differential Approximation.- 15. Conditions for the Invariance of Difference Schemes for the One-dimensional Equations of Gas Dynamics.- 16. Investigation of Properties of the Artificial Viscosity of Invariant Difference Schemes for the One-dimensional Equations of Gas Dynamics.- 17. Conditions for the Invariance of Difference Schemes for the Two-dimensional Equations of Gas Dynamics.- 18. Investigation of Difference Schemes with Time-splitting Using the Theory of Groups.- IV. Appendix.- A.1 Introduction.- A.2 Difference Schemes for the Equation of Propagation.- A.3 Difference Schemes for the Equations of One-dimensional Gas Dynamics.- References.