Approximation of a differential operator by difference operators
An approximation of the differential operator by parameter-dependent operators such that the result of their application to a function is determined by the values of this function on some discrete set of points — a grid — which become more exact as its parameter (mesh, step of the grid) tends to zero.
Let , be a differential operator which converts any function
of a class of functions
into a function
of a linear normed space
. Let
be the domain of definition of the functions in
, and let there be some discrete subset in
— a grid
— which "becomes more dense" as
. Consider the set
of all functions
defined on the grid only and coinciding with
in the points of the grid. A difference operator is defined as any operator
that converts the grid functions in
into functions
in
. One says that the operator
,
, represents an order
approximation to the differential operator
on
if for any function
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as . Occasionally, an approximation is understood to be the equality
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in the sense of some weak convergence. The approximation of a differential operator by difference operators is used for an approximate computation of the function from the table of values
of the function
and for the approximation of a differential equation by difference equations.
There are two principal methods for constructing operators approximating
.
In the first method is defined as the result of applying the differential operator
to a function in
, obtained by some interpolation formula from the grid function
.
The second method is as follows. In the domain of definition of a function
in
one introduces a grid
, and considers the linear space
of grid functions defined on
. The operator
is constructed as the product of two operators: an operator which converts the function
into the grid function
in
, i.e. into a table of approximate values of
, and an operator which extends
from
to the entire domain
. For instance, in order to approximate the differential operator
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one constructs the grid consisting of points
,
,
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and a grid consisting of the points
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The values of the operator at the points
are defined by the equations:
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Thereafter, the definition of is piecewise linearly extended outside
with possible breaks at the points
,
, only.
Let the norm in F be defined by the formula
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Then, on the class of functions with a bounded third derivative, for
and
the operator
represents an order 1, respectively 2, approximation to
. On the class
of functions with bounded second derivatives, the representation is of order 1 only, for any
.
The task of approximating a differential operator by finite-difference operators is sometimes conditionally considered as solved if a method is found for the construction of the grid function
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determined at the points of only, while the task of completing the function
everywhere on
is ignored. In such a case the approximation is defined by considering the space
as normed, and by assuming, for the grid and for the norm, that for any function
, the function
, which coincides with
at the points of
, satisfies the equation
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The operator is understood to be an operator from
in
, and one says that
represents an order
approximation to
on
if, for
,
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In order to construct an operator which is an approximation to
of given order on sufficiently smooth functions, one often replaces each derivative contained in the expression
by its finite-difference approximation, basing oneself on the following fact. For any integers
and for any
,
, in the equation
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it is possible, by using the method of undetermined coefficients and Taylor's formula, to select numbers not depending on
, so that for any function
with
(
) bounded derivatives, an inequality of the type
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where depends only on
and
, is valid. As an example, suppose one constructs an approximating operator for the Laplace operator
,
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if is the closed square
, and
is its interior
. Assume that
where
is a natural number, and construct the grid with points
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which belong to . The points
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then belong to , for integers
and
. Since
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can be approximated with second-order accuracy on a space of sufficiently smooth functions by the finite-difference operator
if one puts, at the points of
:
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where and
are the values of the functions
and
at the point
.
There are also other methods of constructing operators which are approximations to the operator
on the space of solutions
of the differential equation
, and which satisfy additional conditions.
References
[1] | A.F. Filippov, "On stability of difference equations" Dokl. Akad. Nauk SSSR , 100 : 6 (1955) pp. 1045–1048 (In Russian) |
[2] | I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian) |
Comments
The approximation of a differential operator by difference operators is an ingredient for both the approximation of a differential equation by difference equations and for the approximation of a differential boundary value problem by difference boundary value problems, and is therefore extensively treated in the literature on (finite-) difference methods for ordinary and partial differential equations. The references listed below not only provide discretizations of differential equations and boundary value problems, but also the solution of these problems. References [a1]–[a3], [a5], [a6] are introducing textbooks, whereas [a4], [a7], [a8] and [a9] also present more advanced material.
References
[a1] | W.F. Ames, "Numerical methods for partial differential equations" , Nelson , London (1969) |
[a2] | G.E. Forsythe, W.R. Wasow, "Finite difference methods for partial differential equations" , Wiley (1960) |
[a3] | P.R. Garabedian, "Partial differential equations" , Wiley (1964) |
[a4] | S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian) |
[a5] | J.D. Lambert, "Computational methods in ordinary differential equations" , Wiley (1973) |
[a6] | A.R. Mitchell, D.F. Griffiths, "The finite difference method in partial differential equations" , Wiley (1980) |
[a7] | R.D. Richtmeyer, K.W. Morton, "Difference methods for initial value problems" , Wiley (1967) |
[a8] | A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian) |
[a9] | H.J. Stetter, "Analysis of discretization methods for ordinary differential equations" , Springer (1973) |
Approximation of a differential operator by difference operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_of_a_differential_operator_by_difference_operators&oldid=45200