Methods for the Localization of Singularities in Numerical Solutions of Gas Dynamics Problems

As a result of the numerical simulation of multidimensional gas dynamics problems on a computer, the output information is obtained in the form of immense arrays of numerical data. In this connection, there arises the problem of extracting the actually needed information from these arrays; in other words, it is necessary to solve the problem of information compression. In particular, the numerical solution of gas dynamics problems often aims at the information on the solution singularities-the shock waves, contact interfaces, slip lines, etc. Our book is devoted to the development and investigation of accuracy of the algorithms for the localization of such singularities. In addi­ tion, the questions of development of the algorithms for the classification of singularities into several types (on the basis of shock-capturing numerical solutions of two-dimensional gas dynamics problems) are considered for the first time in the monographic literature. For this purpose, some ideas and methods of the modern theory of digital-image processing and of the pattern recognition theory are used. The information obtained at the output of the systems of the singularities classification presented in this book is rich in content, because it contains both physical and geometrical characteristics of recognized objects. Therefore, such "intellectual" systems of information ex­ traction may be used in the expert systems of automated design of aero­ dynamic bodies which meet some optimality requirements. This is, in our opinion, very attractive from the point of view of applications.

1. Introduction and Necessary Notions from the Theory of Difference Schemes for Gas Dynamics Problems.- 1.1. Original Equations. Jump Conditions in the Case of One-Dimensional Gas Flow.- 1.1.1. Divergence and Nondivergence Form of Equations.- 1.1.2. Jump Conditions.- 1.1.3. Riemann Problem.- 1.2. Jump Conditions in the Case of Two-Dimensional Gas Flow.- 1.3. Homogeneous Difference Schemes and Their Differential Approximations.- 1.3.1. Two Approaches to Construction of Schemes for the Computation of Discontinuous Solutions.- 1.3.2. Two Forms of Differential Approximations.- 1.4. On the Applicability of Progressive Wave-Type Solutions of the First Differential Approximation Equations.- 1.4.1. Singular Points of the First Differential Approximation Equations.- 1.4.2. Numerical Study of the Smeared Shock Wave Structure.- 2. Differential Analyzers of Shock Waves in One-Dimensional Gas Flows.- 2.1. An Introductory Example.- 2.2. Existence and Uniqueness of the Smeared Wave Center in the Solution of the System with Artificial Viscosity.- 2.2.1. General Considerations.- 2.2.2. Analysis of Artificial Viscosities.- 2.3. Scheme Viscosity and Smeared Shock Wave Center Existence.- 2.3.1. Relation Between the Divergence Property of Difference Schemes and the First Differential Approximation Divergence Property.- 2.3.2. On the Existence of the Smeared Shock Wave Center.- 2.4. An Analysis of Difference Schemes of Gas Dynamics.- 2.4.2. Second-Order Schemes.- 2.4.3. Practical Realization of the Shock Wave Differential Analyzer Algorithms (With Regard to Sections 2.2 and 2.3).- 2.5. The Application of Differential Analyzers in Problems of Shock Wave Formation.- 2.5.1. An Analysis of the Shock Wave Formation Problem While Using a Uniform Grid.- 2.5.2. Nonuniform Moving Grid Adapting to the Flow.- 3. Differential Analyzers of Shock Waves in Two-Dimensional Gas Dynamic Computations.- 3.1. Method for Investigating the Properties of Curvilinear Shock Front “Smearing”.- 3.1.1. Inequalities in the Zone of Smearing of a Two-Dimensional Shock Wave.- 3.1.2. An Analysis on the Basis of First Differential Approximation.- 3.2. Localization of the Smeared Shock Wave Center in the Case of a Straight Front.- 3.2.1. Analysis on the Basis of Progressive Wave-Type Solutions of the First Differential Approximation Equations.- 3.2.2. Application of the Scheme Viscosity Norm in the Algorithms of Shock Wave Analyzers.- 3.3. Shock Localization by Moving Grids.- 3.3.1. Equations of Inviscid Gas Flow in Moving Coordinates.- 3.3.2. Equations of Grid Motion.- 3.4. Computational Examples.- 4. Differential Analyzers of Contact Discontinuities in One-Dimensional Gas Flows.- 4.1. Methods for the Localization of Contact Discontinuities in the Presence of K-Consistence.- 4.1.1. Investigation Method. Basic Definitions.- 4.1.2. Analysis of Schemes in the Case of the First Differential Approximation K-Consistence.- 4.2. K-Consistence Property of the First Differential Approximation in the Two-Dimensional Case.- 4.3. Methods of K-Inconsistence Suppression.- 4.3.1. Preliminary Discussion.- 4.3.2. Construction of K-Inconsistence Suppression Algorithms in the First-Order Schemes.- 4.3.3. K-Inconsistence Suppression in Second-Order Schemes.- 4.4. The Contact Residual Subtraction Method.- 4.5. Computational Examples.- 5. Optimization Techniques of the Discontinuities Localization.- 5.1. An Analysis of the Miranker—Pironneau Method.- 5.2. Incorporation of the Information on Approximation Viscosity into the Basic Functional.- 5.3. Shock Localization on the Basis of Function Minimization.- 5.4. Gradient Methods of Basic Functional Minimization.- 5.5. Computational Examples.- 5.6. On Optimization Algorithms for the Localization of Contact Discontinuities.- 5.7. An Optimization Method for the Localization of Weak Discontinuities.- 5.8. A Generalization of the Miranker—Pironneau Method for the Case of Polar Coordinates in a Filtration Problem.- 5.8.1. Variational Formulation of a Problem on the Localization of the Saturation Function Discontinuity.- 5.8.2. A Method of Numerical Minimization of the Basic Functional.- 5.8.3. Computational Example.- 6. Difference Solution Refinement in the Neighborhood of Strong Discontinuities.- 6.1. Construction of the Basic Functional.- 6.2. Solution Refinement on the Basis of the Least-Squares Method.- 6.2.1. Formulation of Constrained Optimization Problems.- 6.2.2. Construction of the Discrete Functional and its Minimization.- 6.3. On the Difference Solution Refinement in the Neighborhood of a Shock Wave Front.- 6.4. Computational Examples.- 7. Classification of Singularities in Gas Flows as the Pattern Recognition Problem.- 7.1. Methodologies of Pattern Recognition.- 7.2. Image Formation.- 7.2.1. Image Input.- 7.2.2. Image Preprocessing.- 7.3. Image Segmentation.- 7.3.1. Image Segmentation on a Rectangular Grid.- 7.3.2. Accuracy Assessment of the Image Segmentation Techniques.- 7.3.3. Image Segmentation on a Curvilinear Grid.- 7.4. Feature Space.- 7.5. Algorithms of Pattern Classification.- 7.5.1. Minimum-Distance Classifier.- 7.5.2. Sequential Classification.- 7.5.3. Classification by Feature Functions.- 7.6. Examples of Pattern Classification in the Numerical Solutions of Two-Dimensional Gas Dynamics Problems.- 7.6.1. Tests of Pattern Classification Algorithms.- 7.6.2. Recognition of Shock Waves in Transonic Flow Around an Airfoil.- 7.6.3. Cylindrical Shock Problem.- 7.6.4. Double Mach Reflection of the Strong Shock Wave.- 7.6.5. Supersonic Flow in a Wind Tunnel with a Lower-Wall Step.- Concluding Remarks.- References.