The Method of Differential Approximation

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.

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.- 1.1 Cauchy’s Problem.- 1.2 One-dimensional Time-dependent Case.- 1.3 Systems of Second-order Equations.- 1.4 Basic Concepts of the Theory of Difference Schemes.- 2. The Concept of the Differential Approximation of a Difference Scheme.- 2.1 ?-form and ?-form of the Differential Representation of a Difference Scheme.- 2.2 General Form of the ?-form.- 2.3 ?- and ?-form of the First Differential Approximation.- 2.4 Remarks on Nonlinear Differential Equations.- 2.5 The Role of the First Differential Approximation.- 2.6 On the Correctness of Giving the ?-form as an Infinite Differential Equation.- 2.7 Differential Representations of Difference Schemes in Spaces of Generalized Functions.- 2.8 Asymptotic Expansion of the Solution of a Difference Scheme.- 2.9 On the Injective Character of the Mapping of Difference Schemes in the Set of Differential Representations.- 3. Stability Analysis of Difference Schemes with Constant Coefficients by Means of the Differential Representation.- 3.1 Absolute and Conditional Approximation.- 3.2 Lax’ Equivalence Theorem.- 3.3 On the Necessary Stability Conditions for Difference Schemes.- 4. Connection Between The Stability of Difference Schemes and the Properties of Their First Differential Approximations.- 4.1 Simple Difference Schemes.- 4.2 Majorant Difference Schemes.- 4.3 Fractional-step Method.- 4.4 The Case of Multi-dimensional Schemes.- 4.5 Two-level Difference Schemes.- 4.6 Remarks on Nonlinear Equations.- 5. Dissipative Difference Schemes for Hyperbolic Equations.- 5.1 Different Definitions of Dissipativity.- 5.2 Stability Theorem for Dissipative Schemes in the Generalized Sense.- 5.3 Stability Theorem for Dissipative Schemes in the Sense of Roshdestvenskii-Yanenko-Richtmyer.- 5.4 Stability Theorem for a Partly Dissipative Scheme.- 6. A Means for the Construction of Difference Schemes with Higher Order of Approximation.- 6.1 Convergence Theorem.- 6.2 A Weakly Stable Difference Scheme.- 6.3 Construction of a Third-order Difference Scheme.- 6.4 Application to Nonlinear Equations.- 6.5 Application of the Method to a Boundary Value Problem.- 6.6 Stability Theorems for Dissipative Schemes.- II. Investigation of the Artificial Viscosity of Difference Schemes.- 7. K-property of Difference Schemes.- 7.1 Introduction.- 7.2 Definition of K-property.- 7.3 Simple Difference Schemes.- 7.4 Three-point Schemes.- 7.5 Necessary and Sufficient Conditions for the Strong Property K.- 7.6 Predictor-Corrector Scheme.- 7.7 Implicit Difference Schemes.- 7.8 Higher-order Difference Schemes.- 7.9 Application to Gas Dynamics.- 7.10 Connection Between Partly Dissipative Difference Schemes and Those with the Strong Property K.- 7.11 The Property Pj(p, 1).- 7.12 The Property Dj(p, 1).- 8. Investigation of Dissipation and Dispersion of Difference Schemes.- 8.1 Dissipation and Dispersion of Difference Schemes.- 8.2 Dissipation and Dispersion of Differential Approximations.- 8.3 Relative Dissipative Error and Dispersion.- 8.4 Geometrical Illustration of Dissipative and Dispersive Errors.- 8.5 Classification of Difference Schemes According to Dissipative Properties.- 8.6 Some Remarks on Using Finite Number of Terms of the Differential Approximation.- 8.7 Connection Between Dispersion, Dissipation and Errors of Difference Schemes.- 9. Application of the Method of Differential Approximation to the Investigation of the Effects of Nonlinear Transformations.- 9.1 Introduction.- 9.2 Equivalence of Difference Schemes.- 9.3 The Fluid Equations Including Gravity.- 9.4 The Equations of Gas Dynamics.- 10. Investigation of Monotonicity of Difference Schemes.- 10.1 Introduction.- 10.2 Moving Shock with Constant Velocity.- 11. Difference Schemes in an Arbitrary Curvilinear Coordinate System.- 11.1 Introduction.- 11.2 Definition of a Mesh.- 11.3 Closeness of Solutions of Difference Schemes on Different Meshes.- 11.4 Example of Convective Equation.- III. Invariant Difference Schemes.- 12. Some Basic Concepts of the Theory of Group Properties of Differential Equations.- 12.1 Infinitesimal Operator of Gr.- 12.2 Invariant Subsets of Gr.- 12.3 Necessary and Sufficient Conditions for the Invariance of the First-order Differential Equations.- 13. Groups Admitted by the System of the Equations of Gas Dynamics.- 13.1 Lie-Algebra for Two-dimensional Gas Dynamics.- 13.2 One-dimensional 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.- 15.1 The Class of Two-level Difference Schemes for the Eulerian Equations of Gas Dynamics.- 15.2 Condition for Invariance for the Difference Scheme (15.1).- 15.3 Property M of a Difference Scheine.- 15.4 Property K.- 15.5 Weak Solutions of Difference Scheme (15.1), ? = ?.- 15.6 One-dimensional System of the Equations of Gas Dynamics in Lagrangean Coordinates.- 15.7 Polytropic Gas.- 16. Investigation of Properties of the Artificial Viscosity of Invariant Difference Schemes for the One-dimensional Equations of Gas Dynamics.- 16.1 ?-matrices in Eulerian Coordinates.- 16.2 Property K?.- 16.3 Polynomial Form of the Viscosity Matrix.- 16.4 Numerical Experiments for Equations of Gas Dynamics in Eulerian Coordinates.- 16.5 Damping of Oscillatory Effects.- 16.6 ?-matrices in Lagrangean Coordinates.- 16.7 Width of Shock Smearing.- 16.8 Numerical Experiments on the Equations of Gas Dynamics in Lagrangean Coordinates.- 16.9 Conservative and Fully Conservative Schemes.- 17. Conditions for the Invariance of Difference Schemes for the Two-dimensional Equations of Gas Dynamics.- 17.1 Two-level Class of Difference Schemes.- 17.2 Conditions for the Invariance of the Difference Scheme (17.1).- 17.3 Theorem on Invariance.- 17.4 Property M.- 17.5 Property K?.- 17.6 Some Remarks on Stability.- 17.7 ?-matrices.- 17.8 Numerical Experiments for the Problem of the Interaction of a Shock with Obstacles.- 17.9 Numerical Experiments for the Shallow Water Equations.- 17.10 Analysis of the Properties of the Artificial Viscosity for Difference Schemes of Two-dimensional Gas Flows.- 17.10.1 Schemes of Class J1.- 17.10.2 Schemes of Class J2.- 17.10.3 The Lax’ Scheme.- 17.10.4 The Rusanov-Scheme.- 17.10.5 The Lax-Wendroff-Scheme.- 17.10.6 Two-step Variant of the Lax-Wendroff-Scheme.- 17.10.7 Modification of the Lax-Wendroff-Scheme.- 17.10.8 The MacCormack-Scheme.- 17.10.9 Comparison of Numerical Results.- 18. Investigation of Difference Schemes with Time-splitting Using the Theory of Groups.- 18.1 Two First-order Schemes with Time-splitting.- 18.2 Group Properties of the Schemes (18.1) and (18.2).- 18.3 Conditions for Invariance for a Polytropic Gas.- 18.4 Comparison of Invariant and Noninvariant Schemes.- 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.