The concise look at the Difference Potentials Method and at the causes of new possibilities
which are provided by this method; three examples of solved applied problems and also the notes
about connections between Calderon-Seely and new potentials.
Îáùèé âçãëÿä íà ìåòîä ðàçíîñòíûõ ïîòåíöèàëîâ è íà ïðè÷èíû íîâûõ âîçìîæíîñòåé, êîòîðûå äàåò ýòîò
ìåòîä; òðè ïðèìåðà èç ÷èñëà ðåøåííûõ ïðèêëàäíûõ çàäà÷; çàìå÷àíèÿ î ñâÿçÿõ ìåæäó íîâûìè
ïîòåíöèàëàìè è ïîòåíöèàëàìè Êàëüäåðîíà-Ñèëè.
The method of difference potentials (DPM) is intended for digital
simulation and numerical solution of some problems of mathematical physics DPM
was proposed by the author in his doctoraite (D. Sc.) thesis in 1969 and was
significantly developed at Keldysh Institute Russian Academy of Sciences, at
the Department of Computational Mathematics of the Moscow Institute of Physics
and Technology, at the Institute for Mathematics Modeling of the Russian
Academy of Sciences, at ICASE (NASA Lengley Reserch Center) as well as some
This report will be necessarily
fragmentary. The almost modern state-of-the art of the DPM is reflected in .
There (in ) are also named others
people who take part in developing of DPM. Here I would like to mention only
Professor Semeon Tsynkov, whose participation was very bright and significant.
The new possibilities provided by the
DPM originate from the fact, that DPM combines several advantages of the
classical potentials and Calderon’s-type potentials with the universality and
constructively of difference schemes.
The main advantage of classical
potential method for discretisation and numerical solution the different
problems of mathematical physics (Laplas and Helmholtz equations, the Lame,
Stocks, Maxwell, Cauhy-Riemann systems) is
the possibility to digitize the potentials and boundary integral
equations by means of quadratures. But the kernels of integral operators of
this kind are constructed by means of fundamental solutions, which must be
sufficiently simple. Therefore the field of applicability of classical
potentials method is bounded, to say the least, by equations with constant
coefficients, and can’t be used for
equations with variable coefficients.
A.P.Calderon  and R.T.Seeley 
have constructed and studied the pseudodifferential potentials, boundary
pseudodifferential equations with projectors for elliptic equations with
variable coefficients. But Calderon-Seeley equations can’t be digitized by
means of quadratures, because they do not contain any integrals.
We have constructed some simple
auxiliary boundary problems which are used instead of using symbols of differential
operators. These modified potentials are defined not only for elliptic
equations, they have similar properties, but they also can’t be digitized by
means of quadratures because they also do not contain the integrals. However,
we have constructed earlier the difference potentials. Those can be used for
discrete approximation of new potentials and boundary pseudodifferential
equations with projectors, connected with them. It is possible to say that the
difference potentials play the role of non-classic quadratures for modified
Thus the modified Calderon-Seeley
potentials can be used for discrete simulation and numerical solution of
different problems of mathematical physics, not only elliptic.
Note, however, that really the
difference potentials were proposed much earlier then modification of
Calderon-Seeley potentials was made (1983). This modification was made firstly
to close the constructions of difference potentials when grid step tends to
The difference potentials can be used
also for discrete simulation and numerical solution of different problems
directly, without using the modified Calderon-Seeley potentials.
Note also, that theory of DPM is
mainly algebraic and algorithmic. The general metric properties of new
potentials are studied only partially (see , part I-III). We will speak here
below only about one sort of difference potentials, namely, about the so-called
difference potentials of Cauchy-type and about three examples of applications
of DPM to some numerical problems.
I. The constructions and properties of
Cauchy-type difference potentials.
difference potentials of Cauchy-type and more general potentials are
constructed for solutions of linear system of difference equations of general
form. But we will speak here only about difference potentials of Cauchy-type,
using the Poisson equation and its simplest
is five-point stencil
of difference scheme. Namely, we use grid , with step h; and stencil , which contain following five points:
The auxiliary difference problem.
To construct the
difference potentials we use the following auxiliary difference problems.
Let be a bounded domain
in the xy-plane. We will assume that is some square, whose
sides are lying on grid line x=kh, or y=lh, where k and l are any integers (Fig.1).
Let M0 be the set
of points , which belong to interior part of square (the black points on
Fig.1). We consider the following difference equation
the left-hand side of this equation is meaningful for the functions , whose grid domain is . We add some linear homogeneous condition to the difference
equation (1). For example we assume
We can rewrite
these boundary conditions as inclusion
is the space of all
functions , which equal to zero on boundary .
I. 2. The grid boundary and Cauchy-type potential with jump.
Let be given subdomain of
(Fig.2). By we denote the set of
points m lying in the interior of or on its boundary , and consider the equation
left-hand side of this equation is meaningful only for the functions , defined on the set
By we denote the domain and by the set of grid
points lying in :
the set . The left-hand side of system (4) is meaningful for the
function that are defined on
system (1) splits into two subsystems (3) and (4) whose solutions are defined
on and respectively.
Let us define the boundary g between grid domains and , by setting
We shall speak that
functions from space are regular.
Let and be two arbitrary
regular functions. We define the piecewise-regular function , by setting
us introduce the linear space all functions of the
form (5). The function (5) takes two values and at each point of the grid boundary g.
A single-value function , defined at the points by the formula
be called a jump of the piecewise-regular function on the grid boundary g.
The piecewise-regular function (5) will be called a piecewise-regular solution
of the problem
the functions and satisfy the
be an arbitrary
function which is defined on g.
There exists one and only one piecewise-regular solution of the problem (6)
with jump .
The piecewise-regular solution of problem (6) with
given jump will be called a
difference potential with density .
I. 3. An analogy between the Cauchy-type difference potential and the
classical Cauchy-type integral.
Suppose that G is
a non-self-intersecting closed contour that divides the complex plane z=x+iy into the bounded part and the complimentary
unbounded part . The classical Cauchy-type integral
be determined as a piecewise analytic function tending to zero at infinity and
exhibiting the jump on the contour G.
Here and are the values of the
Cauchy-type integral (7) for and accordingly.
A Cauchy-type integral can be
interpreted as a potential for the solutions of the Cauchy-Riemann system
the real and imaginary parts of an analytic function.
Thus the Cauchy-type
plays the same role for solutions of general linear difference equations
as the Cauchy-type integral plays for solutions of the Cauchy-Riemann system.
II. The examples of applications of DPM.
The difference Cauchy-type potentials
play the same role for general difference schemes as Cauchy-type integral play
for Cauchy-Riemann system. The
Cauchy-type integral has many applications to different problems, which are
connected with Cauchy-Riemann system. Therefore it is naturally to expect, that
the difference Cauchy-type potentials must have many applications to the
different and distant one to other problems. Really, it is so.
We give here several examples of
applications of DPM.
These and many others ones are
reflected in the book  and many papers.
II. 1. Artificial boundary conditions for
A typical example of
problem requiring the construction of artificial boundary conditions is the
problem of calculating the velocity and pressure of the air flow around a body,
usually in the close vicinity of the body. However, for computation we have to
take considerably larger neighborhood.
The DPM gives the
possibilities to construct the nonlocal artificial boundary conditions on the
external boundary of the finite computational subdomain, which demonstrate some
advantages in comparison with ordinary classical artificial boundary
conditions. It was demonstrated by means of some NASA tests (, part V).
II. 2. Nonreflecting
artificial boundary conditions for long-time calculations of acoustic and
electromagnetic waves propagation.
Let we need to calculate a table of
values some function u(t,x,y,z)
on grid of points h>0, which are lying
inside of the unique sphere
on the time-levels and T can be arbitrary great number.
Obviously, the amount of grid points, where we have to find function u(t,x,y,z) has the order of . Therefore there is no method, which requires less then arithmetic operations
to calculate this table.
We consider now this problem in case when function u(t,x,y,z) is the solution of the following
The calculation by means of ordinary difference scheme with grid mech h
for x,y,z and for t requires arithmetic operations
and becomes impossible even for not very large values of T.
But using the lacunas of wave equation
in 3 dimensional space x,y,z together
whit DPM we constructed algorithm which
takes only and which therefore
can’t be improved.
Recently this result was obtained also
for solution Maxwell system,
namely, were constructed non-reflecting artificial boundary conditions for
change the Maxwell system in vacuum out computational subdomain . These results valid also, if inside sphere there are some
scattering, cavities or any non linearity.
These results can be used for
long-time calculations of some problems of acoustic and electrodynamics (see
, part VII).
II. 3. Difference simulating of active shielding problem.
We will consider following problem,
which describes single-frequency harmonic sound
domain (see Fig.2). We will assume that , where
density of useful sound of sources, and function
is a density of noise sources accordingly. We
will assume also, that neither f(x,y)
nor u(x,y) is given. We know only two functions and on boundary , which reflect the sum either the useful sound and noise.
The problem of active shielding the
subdomain against sources,
which are located outside , we are setting in the following way: to construct function g(x,y), , in order that the solution v(x,y) of the problem
the solution w(x,y) of the problem
coinciding on domain :
function g(x,y) in this case will be
called the active suppress of noise.
We will construct now digital simulation of this problem
and give its general solution.
Instead of problem (8) we consider its
five-points difference analogue (Fig.2)
consider the difference analogue of problem (9)
, is an arbitrary grid function.
We also consider the problem
grid function will be called active digital suppress function, if the following
equality is valid:
active suppress digital functions have the form
is an arbitrary
function from which coincides with
the grid boundary g.
Note that the function are unknown. We know
only which can be measured.
And we know, that influence of active suppress function has the same effect
as if we switch off all noise sources, i.e. change , supposing
About the more general
setting and solution of active shielding problem see , part VIII.
III. On some properties of modified Calderon
potentials and approximation of these
potentials by means of difference potentials.
general scheme of modified Calderon-Seeley potentials is based on using of
auxiliary boundary-value problems instead of the symbols of differential
operators. The modified in accordance with this general scheme potentials save
the main properties of Calderon-Seeley potentials and have some additional
useful characteristics. Namely, modified potentials contains some unrestraint
of choice of the auxiliary problems, which can be taken in attention of the
particularity of problem we have to consider. The modified Calderon-Seeley
potential can be approximated by difference potential and then can be used for
numerical solution of the original problem.
The examples of application of DPM
given above and many other ones, which book  contains or reflects, allow to
say that DPM is one among other numerical methods.
At the same time the book  contains mainly the algebraic and
algorithmic parts of theory DPM, but metric part is developed much less.
Maybe it would be interesting to
obtain the theorems about metric characteristics of modified Calderon-Seeley
potentials and accordance pseudo-differential boundary projectors for different
classes of differential equations like that, as it is made for Calderon
potentials and boundary pseudodifferential projectors of elliptic equations by
Maybe it would be interestingly
also to obtain some theorems about metric characteristics of difference
potentials and about approximation of modified Calderon’s potential by means of
difference ones, when mesh step tends to zero.
The example of similar investigation
contains in the part I of book , where
the problems, which are connected with Poisson equation and
corresponding fife point difference scheme, were considered .
V.S. Ryaben’kii, Method of Difference Potentials and Its Applications,
Springer-Verlag, 2002, p.538.
A.P.Calderon (1964), Boundary-value problems of elliptic equations. In:
Proceedings of Sovief-American Symposium in Partial Differential Equations,
Novosibirsk, Fizmatgiz, Moscow, pp.303-304.
R.T.Seeley (1966), Singular integrals and boundary-value problems. Amer. J.
Math., 88, No. 4, pp.781-809.