View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129401.png" />, be a differential operator which converts any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129402.png" /> of a class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129403.png" /> into a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129404.png" /> of a linear normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129405.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129406.png" /> be the domain of definition of the functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129407.png" />, and let there be some discrete subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129408.png" /> — a grid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a0129409.png" /> — which  "becomes more dense"  as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294010.png" />. Consider the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294011.png" /> of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294012.png" /> defined on the grid only and coinciding with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294013.png" /> in the points of the grid. A difference operator is defined as any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294014.png" /> that converts the grid functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294015.png" /> into functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294017.png" />. One says that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294019.png" />, represents an order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294020.png" /> approximation to the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294021.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294022.png" /> if for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294023.png" />

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as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294026.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294028.png" /> from the table of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294029.png" /> of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294030.png" /> and for the [[Approximation of a differential equation by difference equations|approximation of a differential equation by difference equations]].

There are two principal methods for constructing operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294031.png" /> approximating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294032.png" />.

In the first method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294033.png" /> is defined as the result of applying the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294034.png" /> to a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294035.png" />, obtained by some interpolation formula from the grid function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294036.png" />.

The second method is as follows. In the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294037.png" /> of definition of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294039.png" /> one introduces a grid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294040.png" />, and considers the linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294041.png" /> of grid functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294042.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294043.png" /> is constructed as the product of two operators: an operator which converts the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294044.png" /> into the grid function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294046.png" />, i.e. into a table of approximate values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294047.png" />, and an operator which extends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294048.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294049.png" /> to the entire domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294050.png" />. For instance, in order to approximate the differential operator

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one constructs the grid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294052.png" /> consisting of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294054.png" />,

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and a grid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294057.png" /> consisting of the points

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The values of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294060.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294061.png" /> are defined by the equations:

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Thereafter, the definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294064.png" /> is piecewise linearly extended outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294065.png" /> with possible breaks at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294067.png" />, only.

Let the norm in F be defined by the formula

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Then, on the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294069.png" /> with a bounded third derivative, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294071.png" /> the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294072.png" /> represents an order 1, respectively 2, approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294073.png" />. On the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294074.png" /> of functions with bounded second derivatives, the representation is of order 1 only, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294075.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294077.png" /> only, while the task of completing the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294078.png" /> everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294079.png" /> is ignored. In such a case the approximation is defined by considering the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294080.png" /> as normed, and by assuming, for the grid and for the norm, that for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294081.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294082.png" />, which coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294083.png" /> at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294084.png" />, satisfies the equation

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The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294086.png" /> is understood to be an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294087.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294088.png" />, and one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294089.png" /> represents an order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294090.png" /> approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294091.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294092.png" /> if, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294093.png" />,

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In order to construct an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294096.png" /> which is an approximation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294097.png" /> of given order on sufficiently smooth functions, one often replaces each derivative contained in the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294098.png" /> by its finite-difference approximation, basing oneself on the following fact. For any integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a01294099.png" /> and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940101.png" />, in the equation

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it is possible, by using the method of undetermined coefficients and Taylor's formula, to select numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940103.png" /> not depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940104.png" />, so that for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940105.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940106.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940107.png" />) bounded derivatives, an inequality of the type

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940109.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940111.png" />, is valid. As an example, suppose one constructs an approximating operator for the Laplace operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940112.png" />,

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if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940114.png" /> is the closed square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940115.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940116.png" /> is its interior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940117.png" />. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940118.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940119.png" /> is a natural number, and construct the grid with points

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which belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940121.png" />. The points

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then belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940123.png" />, for integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940124.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940125.png" />. Since

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940126.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940127.png" /> can be approximated with second-order accuracy on a space of sufficiently smooth functions by the finite-difference operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940128.png" /> if one puts, at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940129.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940130.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940131.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940132.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940133.png" /> are the values of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940134.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940135.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940136.png" />.

There are also other methods of constructing operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940137.png" /> which are approximations to the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940138.png" /> on the space of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940139.png" /> of the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012940/a012940140.png" />, and which satisfy additional conditions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.F. Filippov,   "On stability of difference equations"  ''Dokl. Akad. Nauk SSSR'' , '''100''' :  6  (1955)  pp. 1045–1048  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.S. Berezin,   N.P. Zhidkov,   "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR></table>



====Comments====
The approximation of a differential operator by difference operators is an ingredient for both the [[Approximation of a differential equation by difference equations|approximation of a differential equation by difference equations]] and for the [[Approximation of a differential boundary value problem by difference boundary value problems|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 [[#References|[a1]]]–[[#References|[a3]]], [[#References|[a5]]], [[#References|[a6]]] are introducing textbooks, whereas [[#References|[a4]]], [[#References|[a7]]], [[#References|[a8]]] and [[#References|[a9]]] also present more advanced material.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.F. Ames,   "Numerical methods for partial differential equations" , Nelson , London  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.E. Forsythe,   W.R. Wasow,   "Finite difference methods for partial differential equations" , Wiley  (1960)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1964)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.K. Godunov,   V.S. Ryaben'kii,   "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.D. Lambert,   "Computational methods in ordinary differential equations" , Wiley  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.R. Mitchell,   D.F. Griffiths,   "The finite difference method in partial differential equations" , Wiley  (1980)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.D. Richtmeyer,   K.W. Morton,   "Difference methods for initial value problems" , Wiley  (1967)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.A. Samarskii,   "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  H.J. Stetter,   "Analysis of discretization methods for ordinary differential equations" , Springer  (1973)</TD></TR></table>