Ebook: Method of Difference Potentials and Its Applications

The method of difference potentials (MDP) was proposed in [1]-[8] and sig­ nificantly developed in [9]-[101] and some other works. The present book describes the current state of the art in the method of difference potentials and is a revised and essentially supplemented version of the author's first book devoted to this method, which was published by "Nauka" in 1987 [100]. This monograph deals with the MDP apparatus and several of its appli­ cations, particularly to the following problems: 1. the numerical solution ofinterior and exterior boundary-value problems for systems of partial differential equations; 2. the construction of conditions at the artificial boundary ofthe compu­ tational domain, which equivalently replace the equations and conditions at infinity in stationary problems of gas flowpast immersed bodies as well as in some other steady-state problems; 3. the spectral approach to the construction of artificial boundary con­ ditions replacing the equations of propagation of physical fields outside the computational domain containing perturbation sources; 4. the construction of artificial boundary conditions on the boundary of the computational domain for numerically solving the scattering problems in large time in a neighborhood of a fixed or a moving scatterer; 5. the statement and solution of stationary mathematical problems of the active shielding of a given subdomain from the influence of perturbation sources located outside the screened subdomain.

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The book presents the method of difference potentials first proposed by the author in 1969 and contains illustrative examples and new algorithms for solving applied problems of gas dynamics, diffraction, scattering theory, and active noise screening. The fundamentals of the method are described in Parts I-III and its applications in Parts IV-VIII. To get acquainted with the basic ideas of the method, it suffices to study the Introduction. After this, each of the Parts VI-VIII can be read independently. The book is intended for specialists in the field of computational mathematics and the theory of differential and integral equations, as well as for graduate students of related specialities.

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The book presents the method of difference potentials first proposed by the author in 1969 and contains illustrative examples and new algorithms for solving applied problems of gas dynamics, diffraction, scattering theory, and active noise screening. The fundamentals of the method are described in Parts I-III and its applications in Parts IV-VIII. To get acquainted with the basic ideas of the method, it suffices to study the Introduction. After this, each of the Parts VI-VIII can be read independently. The book is intended for specialists in the field of computational mathematics and the theory of differential and integral equations, as well as for graduate students of related specialities.
Content:
Front Matter....Pages I-XVIII
Introduction....Pages 1-32
Front Matter....Pages 33-35
Preliminaries....Pages 37-52
Differential and Difference Potentials....Pages 53-80
Reduction of Boundary-Value Problems for the Laplace Equation to Boundary Equations of Calder?n—Seeley Type....Pages 81-86
Numerical Solution of Boundary-Value Problems....Pages 87-136
Front Matter....Pages 137-139
Generalized Potentials and Boundary Equations with Projections for Differential Operators....Pages 141-158
General Constructions of Potentials and Boundary Equations for Difference Operators....Pages 159-206
Lazarev’s Results on the Algebraic Structure of the Set of Surface Potentials of a Linear Operator....Pages 207-212
Front Matter....Pages 213-215
A General Scheme of the Method of Difference Potentials for Differential Problems....Pages 217-272
Illustrations of Constructions of the Method of Difference Potentials....Pages 273-290
General Scheme of the Method of Difference Potentials for Solving Numerically the Difference Analogs of Differential Boundary-Value Problems....Pages 291-324
Front Matter....Pages 325-327
The Tricomi Problem....Pages 329-340
Constructions of the Method of Difference Potentials for the Computation of Stressed States of Elastic Compressible Materials....Pages 341-344
Problems of Internal Flows of Viscous Incompressible Fluids....Pages 345-370
An Example of the MDP Algorithm for Computing the Stationary Acoustic Wave Field outside a Solid of Revolution....Pages 371-390
Front Matter....Pages 391-394
An Efficient Algorithm for Constructing Artificial Boundary Conditions for a Model Problem....Pages 395-402
On the Results of the Application of the Method of Difference Potentials to the Construction of Artificial Boundary Conditions for External Flow Computations....Pages 403-441
Front Matter....Pages 443-443
Nonreflecting Difference Conditions on the Moving and Shape Varying Boundary of the Computational Domain....Pages 445-460
Spectral Approach to the Construction of Nonreflecting Boundary Conditions....Pages 461-484
Front Matter....Pages 485-487
Problem of Constructing NRABCs and the Corresponding Auxiliary Cauchy Problem....Pages 489-494
Front Matter....Pages 485-487
Algorithm for Solving the Cauchy Problem with the Help of Lacunas....Pages 495-510
Front Matter....Pages 511-513
Active Shielding Control....Pages 515-518
Difference Imitation Problems....Pages 519-522
Back Matter....Pages 523-538

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The book presents the method of difference potentials first proposed by the author in 1969 and contains illustrative examples and new algorithms for solving applied problems of gas dynamics, diffraction, scattering theory, and active noise screening. The fundamentals of the method are described in Parts I-III and its applications in Parts IV-VIII. To get acquainted with the basic ideas of the method, it suffices to study the Introduction. After this, each of the Parts VI-VIII can be read independently. The book is intended for specialists in the field of computational mathematics and the theory of differential and integral equations, as well as for graduate students of related specialities.
Content:
Front Matter....Pages I-XVIII
Introduction....Pages 1-32
Front Matter....Pages 33-35
Preliminaries....Pages 37-52
Differential and Difference Potentials....Pages 53-80
Reduction of Boundary-Value Problems for the Laplace Equation to Boundary Equations of Calder?n—Seeley Type....Pages 81-86
Numerical Solution of Boundary-Value Problems....Pages 87-136
Front Matter....Pages 137-139
Generalized Potentials and Boundary Equations with Projections for Differential Operators....Pages 141-158
General Constructions of Potentials and Boundary Equations for Difference Operators....Pages 159-206
Lazarev’s Results on the Algebraic Structure of the Set of Surface Potentials of a Linear Operator....Pages 207-212
Front Matter....Pages 213-215
A General Scheme of the Method of Difference Potentials for Differential Problems....Pages 217-272
Illustrations of Constructions of the Method of Difference Potentials....Pages 273-290
General Scheme of the Method of Difference Potentials for Solving Numerically the Difference Analogs of Differential Boundary-Value Problems....Pages 291-324
Front Matter....Pages 325-327
The Tricomi Problem....Pages 329-340
Constructions of the Method of Difference Potentials for the Computation of Stressed States of Elastic Compressible Materials....Pages 341-344
Problems of Internal Flows of Viscous Incompressible Fluids....Pages 345-370
An Example of the MDP Algorithm for Computing the Stationary Acoustic Wave Field outside a Solid of Revolution....Pages 371-390
Front Matter....Pages 391-394
An Efficient Algorithm for Constructing Artificial Boundary Conditions for a Model Problem....Pages 395-402
On the Results of the Application of the Method of Difference Potentials to the Construction of Artificial Boundary Conditions for External Flow Computations....Pages 403-441
Front Matter....Pages 443-443
Nonreflecting Difference Conditions on the Moving and Shape Varying Boundary of the Computational Domain....Pages 445-460
Spectral Approach to the Construction of Nonreflecting Boundary Conditions....Pages 461-484
Front Matter....Pages 485-487
Problem of Constructing NRABCs and the Corresponding Auxiliary Cauchy Problem....Pages 489-494
Front Matter....Pages 485-487
Algorithm for Solving the Cauchy Problem with the Help of Lacunas....Pages 495-510
Front Matter....Pages 511-513
Active Shielding Control....Pages 515-518
Difference Imitation Problems....Pages 519-522
Back Matter....Pages 523-538
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