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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Introduction

Abstract
The method of difference potentials is intended for digital simulation of problems of mathematical physics. In particular, one of the most important applications of this method is the numerical solution of interior and exterior boundary-value problems for linear partial differential equations. In this traditional area of computational mathematics, the most frequently used methods are the finite-difference method and the boundary elements method.
Viktor S. Ryaben’kii

### 1. Preliminaries

Abstract
Here we recall some well-known definitions and facts to be used later. A first reading can be restricted to the first three sections.
Viktor S. Ryaben’kii

### 2. Differential and Difference Potentials

Abstract
In this chapter, we describe and discuss differential potentials for the Laplace operator, their various interpretations and representations, and, particularly, their operator form resembling the one studied by Calderon [109] and Seeley [110] for general elliptic operators. Furthermore, we construct a difference potential for the difference analog of the Laplace operator and present a method for approximating a differential potential with a given density by a difference potential, which is relatively easy to calculate. We have used the term “differential” for potentials associated with differential operators so as to distinguish them from “difference” potentials corresponding to difference operators.
Viktor S. Ryaben’kii

### 3. Reduction of Boundary-Value Problems for the Laplace Equation to Boundary Equations of Calderón—Seeley Type

Abstract
It will be shown in this chapter how to replace the Laplace equation in a domain by equivalent relations connecting the values of the desired solution $$u|_\Gamma$$ and its normal derivative $$\frac{{\partial u}} {{\partial n}}\left| {_\Gamma } \right.$$ on the boundary of the domain. These relations, together with the boundary conditions defining the boundary-value problem, form a system of equations for functions defined only on the boundary equivalent to the original boundary-value problem for the Laplace equation in the domain.
Viktor S. Ryaben’kii

### 4. Numerical Solution of Boundary-Value Problems

Abstract
In this chapter we describe and formulate an algorithm embodying the method of difference potentials for the numerical solution of the boundary-value problems
$$\Delta u_{\overline D } = 0,x \in D,u_{\overline D } \left| {_\Gamma = \phi (s)} \right.,$$
(I)
and
$$\Delta u_{\overline D } = 0,x \in D,\left( {u_D + \frac{{\partial u_D }} {{\partial n}}} \right)\left| {_\Gamma = \phi (s)} \right..$$
(II)
Viktor S. Ryaben’kii

### 1. Generalized Potentials and Boundary Equations with Projections for Differential Operators

Abstract
In this chapter we introduce the notion of a clear trace and define surface potentials with density from the space of clear traces. For differential equations in a domain, we construct equivalent equations on the boundary of this domain, which must be solved for the unknown density from the space of clear traces. These boundary equations contain a projection in their structure. They are a generalization of the classical Sokhotskii-Plemelj conditions in the theory of analytic functions (of the Cauchy-Riemann equations) and of the Calderon-Seeley boundary equations [109, 110].
Viktor S. Ryaben’kii

### 2. General Constructions of Potentials and Boundary Equations for Difference Operators

Abstract
Difference potentials, the related Green formula, a multilayer grid boundary, and boundary equations with projections that are equivalent to the original system of difference equations in a grid domain were first constructed in [1, 2].
Viktor S. Ryaben’kii

### 3. Lazarev’s Results on the Algebraic Structure of the Set of Surface Potentials of a Linear Operator

Abstract
In Chap. 1 we proposed a construction of the potential for a given differential operator with density from the space of clear traces and we constructed the boundary equations for the density, which isolate the traces of solutions of the homogeneous differential equation. We saw that the choice of the Green operator used in the definition of the potential
$$P_{\overline D \Gamma } \xi _\Gamma = \upsilon _{\overline D } - G_{\overline D D} L_{\overline D D} \upsilon _{\overline D }$$
with density $${{\xi }_{\Gamma }} = T{{r}_{\Gamma }}_{{\bar{D}}}{{\upsilon }_{{\bar{D}}}}$$ as well as the choice of the space ΞГ of clear traces and that of the clear trace operator
$${\text{T}}{{r}_{{\Gamma \bar{D}}}}:V_{{\bar{D}}}^{ + } \to {{\Xi }_{\Gamma }}$$
is significantly nonunique. The structure of the set of all potentials was not studied in Chap. 1.
Viktor S. Ryaben’kii

### 1. A General Scheme of the Method of Difference Potentials for Differential Problems

Abstract
The scheme for applying the method of difference potentials (MDP) to problems of the form (I), (II), which will be described in this chapter, is a generalization of the scheme that was outlined in the introduction to the book and then realized in detail in Part I for the main boundary-value problems in the case of the Laplace equation on the plane.
Viktor S. Ryaben’kii

### 2. Illustrations of Constructions of the Method of Difference Potentials

Abstract
We illustrate the basic constructions of the method of difference potentials for problems in a bounded domain, outside a bounded domain, and in a bounded domain with a cut. We also consider an example of boundary equations with projections for the Stokes system.
Viktor S. Ryaben’kii

### 3. General Scheme of the Method of Difference Potentials for Solving Numerically the Difference Analogs of Differential Boundary-Value Problems

Abstract
To solve the differential boundary-value problem numerically, one can first approximate this problem by a difference one and then calculate its solution. In this chapter we present the general scheme of the method of difference potentials used for solving numerically the difference boundary-value problem under study.
Viktor S. Ryaben’kii

### 1. The Tricomi Problem

Abstract
In this chapter we use the method of difference potentials for computing solutions of some difference analogs of the Tricomi problem. The classical Tricomi problem is a typical example of numerous problems arising in the study of plane transonic flows of compressible gas, e.g., of flows in the Laval nozzle or flows around contours. The Tricomi equation has the form
$$y\frac{{\partial ^2 u}} {{\partial x^2 }} + \frac{{\partial ^2 u}} {{\partial y}} = 0.$$
(I)
It is an equation of mixed type, since it is elliptic for y > 0 and hyperbolic for y < 0.
Viktor S. Ryaben’kii

### 2. Constructions of the Method of Difference Potentials for the Computation of Stressed States of Elastic Compressible Materials

Abstract
In this chapter we specialize the difference potential and other main constructions of the method of difference potentials to the Lamé system in a bounded domain D:
$$L_{D\overline D } u_{\overline D } = \mu \Delta u_{\overline D } + (\lambda + \mu ){\text{grad div }}u_{\overline D } = f_D ,$$
(I)
which describes the plane stressed state of isotropic elastic materials. The Lamé constants λ and μ are positive and characterize the elastic properties of the material. The Lamé system is strongly elliptic in the sense of Vishik. The vector function $$u_{\overline D } \left( {x,y} \right) = \left\{ {u_{\overline D }^{(x)} ,u_{\overline D }^{(y)} } \right\}$$ describes displacement from the equilibrium position (x, y).
Viktor S. Ryaben’kii

### 3. Problems of Internal Flows of Viscous Incompressible Fluids

Abstract
Nonstationary flows of viscous incompressible fluids can be modeled as boundary-value problems for systems of differential Navier-Stokes equations. In the case of three-dimensional problems, the desired functions are velocity and pressure. In the case of plane flows, one can also use the stream function and vorticity as desired functions.
Viktor S. Ryaben’kii

### 4. An Example of the MDP Algorithm for Computing the Stationary Acoustic Wave Field outside a Solid of Revolution

Abstract
The steady-state acoustic wave field in a homogeneous medium is governed by the Helmholtz equation
$$\Delta u + \mu ^2 u = 0.$$
(4.1)
Viktor S. Ryaben’kii

### 1. An Efficient Algorithm for Constructing Artificial Boundary Conditions for a Model Problem

Abstract
Consider the following problem with respect to the function u(x, y) defined on the whole plane
$$\Psi u = f\left( {x,y} \right),$$
(1.1)
$$|u\left( {x,y} \right)|{\text{ }} < {\text{ const}}{\text{.}}$$
(1.2)
.
Viktor S. Ryaben’kii

### 2. On the Results of the Application of the Method of Difference Potentials to the Construction of Artificial Boundary Conditions for External Flow Computations

Abstract
Let us first briefly repeat the general arguments behind constructing the artificial boundary conditions (ABCs) for the numerical solution of problems formulated on unbounded domains. As has been mentioned, a standard approach to solving infinite-domain boundary-value problems on the computer involves truncation as a first step, prior to the discretization of the continuous problem and solution of the resulting discrete system. The truncated problem is clearly subdefinite unless supplemented by the proper closing procedure at the external boundary of the finite computational domain. The latter boundary is often called artificial emphasizing the fact that it originates from the numerical limitations rather than original physical formulation. The corresponding closing procedure is called the ABCs.
Viktor S. Ryaben’kii

### 1. Nonreflecting Difference Conditions on the Moving and Shape Varying Boundary of the Computational Domain

Abstract
There is a wide class of unsteady initial boundary-value problems formulated either on the entire Euclidean space R or a large domain D (with boundary conditions), for which the solution u(t, x) needs to be known only on a bounded subdomain of the original domain. Without much loss of generality, we additionally assume that the governing differential equations outside this computational subdomain, as well as boundary conditions at infinity or distant physical boundary, are linear and homogeneous.
Viktor S. Ryaben’kii

### 2. Spectral Approach to the Construction of Nonreflecting Boundary Conditions

Abstract
We consider a family of time-dependent problems with solutions u(x, t) defined for t ≥ 0 in a domain D consisting of the whole x-plane excluding possibly a finite number of subdomains different for each problem, but lying inside the disk
$$S_R = \left\{ {x_1^2 + x_2^2 \leqslant R^2 } \right\}{\text{ for }}R < 1.$$
.
Viktor S. Ryaben’kii

### 1. Problem of Constructing NRABCs and the Corresponding Auxiliary Cauchy Problem

Abstract
Here we introduce the concept of nonreflecting artificial boundary conditions (NRABCs) and reduce the computation of NRABCs on the current time level to solving numerically the auxiliary Cauchy problem for the corresponding difference wave equation.
Viktor S. Ryaben’kii

### 2. Algorithm for Solving the Cauchy Problem with the Help of Lacunas

Abstract
We shall construct an algorithm for solving the difference Cauchy problem in a bounded computational subdomain (for definiteness, in the unit ball) so as to use the lacunas of solutions of the wave equation and of its difference analog in the case of three space variables. The number of arithmetical operations in this algorithm required for passing to the next time level is minimal and does not increase with the number of the level. The results contained in this chapter were obtained in [88].
Viktor S. Ryaben’kii

### 1. Active Shielding Control

Abstract
Let M be an arbitrary finite or countable set of points (a grid). Suppose that to each point mM there corresponds a finite set of points N m , mM, (the stencil of the difference scheme). Let $$\phi \left( {m,U_{N_m } } \right) = \phi \left( {m,U_n } \right),n \in N_m ,$$, be a given function defined for each mM.
Viktor S. Ryaben’kii

### 2. Difference Imitation Problems

Abstract
We shall consider two difference schemes. We assume that the grid domains of definition of solutions to these schemes have the common part N +.
Viktor S. Ryaben’kii

### Backmatter

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