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

as . Occasionally, an approximation is understood to be the equality

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

one constructs the grid consisting of points , ,

and a grid consisting of the points

The values of the operator at the points are defined by the equations:

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

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

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

The operator is understood to be an operator from in , and one says that represents an order approximation to on if, for ,

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

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

where depends only on and , is valid. As an example, suppose one constructs an approximating operator for the Laplace operator ,

if is the closed square , and is its interior . Assume that where is a natural number, and construct the grid with points

which belong to . The points

then belong to , for integers and . Since

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 :

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)