The Method of Differential Approximation

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.