Method of Difference Potentials and Its Applications

ISBN-13: 9783642627156 / Angielski / Miękka / 2012 / 538 str.

Viktor S. Ryaben'kii; Viktor S. Ryaben'kii; N. K. Kulman

Opis

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.

Spis treści

I. Justification of Algorithms of the Method.- 1. Preliminaries.- 2. Differential and Difference Potentials.- 3. Reduction of Boundary-Value Problems for the Laplace Equation to Boundary Equations of Calderón-Seeley Type.- 4. Numerical Solution of Boundary-Value Problems.- II. General Constructions of Surface Potentials.- 1. Generalized Potentials and Boundary Equations with Projections for Differential Operators.- 2. General Constructions of Potentials and Boundary Equations for Difference Operators.- 3. Lazarev’s Results on the Algebraic Structure of the Set of Surface Potentials of a Linear Operator.- III. A General Scheme of the Method of Difference.- 1. A General Scheme of the Method of Difference Potentials for Differential Problems.- 2. Illustrations of Constructions of the Method of Difference Potentials.- 3. General Scheine of the Method of Difference Potentials for Solving Numerically the Difference Analogs of Differential Boundary-Value Problems.- IV. Examples of MDP Algorithms for Solving.- 1. The Tricomi Problem.- 2. Constructions of the Method of Difference Potentials for the Computation of Stressed States of Elastic Compressible Materials.- 3. Problems of Internal Flows of Viscous Incompressible Fluids.- 4. An Example of the MDP Algorithm for Computing the Stationary Acoustic Wave Field outside a Solid of Revolution.- V. Artificial Boundary Conditions.- 1. An Efficient Algorithm for Constructing Artificial Boundary Conditions for a Model Problem.- 2. On the Results of the Application of the Method of Difference Potentials to the Construction of Artificial Boundary Conditions for External Flow Computations.- VI. General Constructions of Difference.- 1. Nonreflecting Difference Conditions on the Moving and Shape Varying Boundary of the Computational Domain.- 2. Spectral Approach to the Construction of Nonreflecting Boundary Conditions.- VII. Nonreflecting Artificial Boundary Conditions.- 1. Problem of Constructing NRABCs and the Corresponding Auxiliary Cauchy Problem.- 2. Algorithm for Solving the Cauchy Problem with the Help of Lacunas.- VIII. Problems of Active Shielding and Imitation.- 1. Active Shielding Control.- 2. Difference Imitation Problems.- References.