Abstract
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 CalderonSeely and new potentials.
Àííîòàöèÿ
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 ÊàëüäåðîíàÑèëè.
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
other Institutes.
This report will be necessarily
fragmentary. The almost modern stateofthe art of the DPM is reflected in [1].
There (in [1]) 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’stype 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, CauhyRiemann 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 [2] and R.T.Seeley [3]
have constructed and studied the pseudodifferential potentials, boundary
pseudodifferential equations with projectors for elliptic equations with
variable coefficients. But CalderonSeeley 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 nonclassic quadratures for modified
CalderonSeeley potentials.
Thus the modified CalderonSeeley
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
CalderonSeeley potentials was made (1983). This modification was made firstly
to close the constructions of difference potentials when grid step tends to
zero.
The difference potentials can be used
also for discrete simulation and numerical solution of different problems
directly, without using the modified CalderonSeeley potentials.
Note also, that theory of DPM is
mainly algebraic and algorithmic. The general metric properties of new
potentials are studied only partially (see [1], part IIII). We will speak here
below only about one sort of difference potentials, namely, about the socalled
difference potentials of Cauchytype and about three examples of applications
of DPM to some numerical problems.
I. The constructions and properties of
Cauchytype difference potentials.
The
difference potentials of Cauchytype 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 Cauchytype,
using the Poisson equation _{} and its simplest
difference analogue
_{}
where
_{} is fivepoint stencil
of difference scheme. Namely, we use grid _{}, with step h; _{} and stencil _{}, which contain following five points:
_{}
I. 1.
The auxiliary difference problem.
To construct the
difference potentials we use the following auxiliary difference problems.
Let _{} be a bounded domain
in the xyplane. 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 M^{0 }be the set
of points _{}, which belong to interior part of square _{} (the black points on
Fig.1). We consider the following difference equation
_{} (1)
Obviously,
the lefthand 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
_{}
Fig.1.
We can rewrite
these boundary conditions as inclusion
_{} (2)
where
_{} is the space of all
functions _{}, which equal to zero on boundary _{}.
I. 2. The grid boundary and Cauchytype 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
_{} (3)
Fig.2.
The
lefthand 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 _{}:
_{}
We
consider system
_{} (4)
on
the set _{}. The lefthand side of system (4) is meaningful for the
function _{} that are defined on
the set
_{}
Thus
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
_{} (see Fig.3).
We shall speak that
functions _{} from space _{} are regular.
Let _{} and _{} be two arbitrary
regular functions. We define the piecewiseregular function _{}, by setting
_{} (5)
Let
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.
Fig.3.
A singlevalue function _{}, defined at the points _{} by the formula
_{}
will
be called a jump of the piecewiseregular function _{} on the grid boundary g.
The piecewiseregular function (5) will be called a piecewiseregular solution
of the problem
_{} (6)
if
the functions _{} and _{} satisfy the
homogeneous equations
_{}
_{}
Theorem. Let
_{} be an arbitrary
function which is defined on g.
There exists one and only one piecewiseregular solution _{} of the problem (6)
with jump _{}.
Definition.
The piecewiseregular solution _{} of problem (6) with
given jump _{} will be called a
difference potential _{} with density _{}.
I. 3. An analogy between the Cauchytype difference potential and the
classical Cauchytype integral.
Suppose that G is
a nonselfintersecting closed contour that divides the complex plane z=x+iy into the bounded part _{} and the complimentary
unbounded part _{}. The classical Cauchytype integral
_{} (7)
can
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
Cauchytype integral (7) for _{} and _{} accordingly.
A Cauchytype integral can be
interpreted as a potential for the solutions of the CauchyRiemann system
_{}
connecting
the real and imaginary parts of an analytic function.
Thus the Cauchytype
difference potentials
_{} plays the same role for solutions of general linear difference equations
as the Cauchytype integral plays for solutions of the CauchyRiemann system.
II. The examples of applications of DPM.
The difference Cauchytype potentials
play the same role for general difference schemes as Cauchytype integral play
for CauchyRiemann system. The
Cauchytype integral has many applications to different problems, which are
connected with CauchyRiemann system. Therefore it is naturally to expect, that
the difference Cauchytype 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 [1] and many papers.
II. 1. Artificial boundary conditions for
stationary problems.
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 ([1], part V).
II. 2. Nonreflecting
artificial boundary conditions for longtime 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 timelevels _{} 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
problem
_{}
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 nonreflecting 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
longtime calculations of some problems of acoustic and electrodynamics (see
[1], part VII).
II. 3. Difference simulating of active shielding problem.
We will consider following problem,
which describes singlefrequency harmonic sound
_{} (8)
on
domain_{} (see Fig.2). We will assume that _{}, where
_{}
is a
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
_{} (9)
and
the solution w(x,y) of the problem
_{} (10)
were
coinciding on domain _{}:
_{}
The
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
fivepoints difference analogue (Fig.2)
_{} (11)
We
consider the difference analogue of problem (9)
_{} (12)
where
_{}, is an arbitrary grid function.
We also consider the problem
_{}
where
_{}
The
grid function will be called active digital suppress function, if the following
equality is valid:
_{}
Theorem. All
active suppress digital functions _{} have the form
_{} (13)
where
_{} is an arbitrary
function from _{} which coincides with
given _{}:
_{} (14)
on
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 [1], part VIII.
III. On some properties of modified Calderon
potentials and approximation of these
potentials by means of difference potentials.
The
general scheme of modified CalderonSeeley potentials is based on using of
auxiliary boundaryvalue problems instead of the symbols of differential
operators. The modified in accordance with this general scheme potentials save
the main properties of CalderonSeeley 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 CalderonSeeley
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 [1] contains or reflects, allow to
say that DPM is one among other numerical methods.
At the same time the book [1] 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 CalderonSeeley
potentials and accordance pseudodifferential 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
Seeley [3].
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 [1], where
the problems, which are connected with Poisson equation and
corresponding fife point difference scheme, were considered .
REFERENCES
[1]
V.S. Ryaben’kii, Method of Difference Potentials and Its Applications,
SpringerVerlag, 2002, p.538.
[2]
A.P.Calderon (1964), Boundaryvalue problems of elliptic equations. In:
Proceedings of SoviefAmerican Symposium in Partial Differential Equations,
Novosibirsk, Fizmatgiz, Moscow, pp.303304.
[3]
R.T.Seeley (1966), Singular integrals and boundaryvalue problems. Amer. J.
Math., 88, No. 4, pp.781809.
