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 $ L, Lu = f $, be a differential operator which converts any function $ u $ of a class of functions $ U $ into a function $ f $ of a linear normed space $ F $. Let $ D _ {U} $ be the domain of definition of the functions in $ U $, and let there be some discrete subset in $ D _ {U} $— a grid $ D _ {hU } $— which "becomes more dense" as $ h \rightarrow 0 $. Consider the set $ U _ {h} $ of all functions $ [u] _ {h} $ defined on the grid only and coinciding with $ u $ in the points of the grid. A difference operator is defined as any operator $ L _ {h} $ that converts the grid functions in $ U _ {h} $ into functions $ f _ {h} $ in $ F $. One says that the operator $ L _ {h} $, $ L _ {h} [u] _ {h} = f _ {h} $, represents an order $ p $ approximation to the differential operator $ L $ on $ U $ if for any function $ u \in U $

$$ \| Lu - L _ {h} [ u ] _ {h} \| _ {F} \rightarrow 0 , $$

$$ \| Lu - L _ {h} [ u ] _ {h} \| _ {F } \leq ch ^ {p} ,\ c = c ( u ) = \textrm{ const } , $$

as $ h \rightarrow 0 $. Occasionally, an approximation is understood to be the equality

$$ \lim\limits _ {h \rightarrow 0 } \ L _ {h} [ u ] _ {h} = Lu $$

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 $ Lu $ from the table of values $ [u] _ {h} $ of the function $ u $ and for the approximation of a differential equation by difference equations.

There are two principal methods for constructing operators $ L _ {h} $ approximating $ L $.

In the first method $ L _ {h} [u] _ {h} $ is defined as the result of applying the differential operator $ L $ to a function in $ U $, obtained by some interpolation formula from the grid function $ [u] _ {h} $.

The second method is as follows. In the domain $ D _ {F} $ of definition of a function $ f $ in $ F $ one introduces a grid $ D _ {hF } $, and considers the linear space $ F _ {h} $ of grid functions defined on $ D _ {hF} $. The operator $ L _ {h} [u] _ {h} $ is constructed as the product of two operators: an operator which converts the function $ [u] _ {h} $ into the grid function $ f _ {h} $ in $ F _ {h} $, i.e. into a table of approximate values of $ f $, and an operator which extends $ f _ {h} $ from $ D _ {hF } $ to the entire domain $ D _ {f} $. For instance, in order to approximate the differential operator

$$ L = \frac{d}{dx} ,\ \frac{du}{dx} = f (x),\ \ 0 \leq z \leq 1 , $$

one constructs the grid $ D _ {hU } $ consisting of points $ x _ {k} $, $ k = 0 \dots N $,

$$ 0 = x _ {0} < \dots < x _ {k} < x _ {k+1} < \dots < x _ {N} = 1 , $$

$$ \max _ { k } ( x _ {k + 1 } - x _ {k} ) = h , $$

and a grid $ D _ {hF } $ consisting of the points

$$ x _ {k} ^ {*} = x _ {k} + \theta ( x _ {k + 1 } - x _ {k} ) ,\ k = 0 \dots N - 1 , $$

$$ 0 \leq \theta \leq 1,\ \theta = \textrm{ const } . $$

The values of the operator $ L _ {h} [u] _ {h} $ at the points $ x _ {k} ^ {*} $ are defined by the equations:

$$ \left . L _ {h} [ u ] _ {h} \right | _ {x = x _ {k} ^ {*} } = \ f _ {h} ( x _ {k} ^ {*} ) = \frac{u ( x _ {k+1} ) -u ( x _ {k} ) }{x _ {k+1} - x _ {k} } , $$

$$ k = 0 \dots N - 1 . $$

Thereafter, the definition of $ L _ {h} [u] _ {h} $ is piecewise linearly extended outside $ D _ {hF } $ with possible breaks at the points $ x _ {k} ^ {*} $, $ k = 1 \dots N - 2 $, only.

Let the norm in F be defined by the formula

$$ \| \phi \| _ {F} = \sup _ { x } | \phi (x) | . $$

Then, on the class of functions $ U $ with a bounded third derivative, for $ \theta = 0 $ and $ \theta = h/2 $ the operator $ L _ {h} $ represents an order 1, respectively 2, approximation to $ L = d/dx $. On the class $ U $ of functions with bounded second derivatives, the representation is of order 1 only, for any $ \theta \in [0, 1] $.

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

$$ \left . L _ {h} [ u ] _ {h} \right | _ {D _ {hF} } = f _ {h} \in F _ {h} , $$

determined at the points of $ D _ {hF } $ only, while the task of completing the function $ f _ {h} $ everywhere on $ D _ {F} $ is ignored. In such a case the approximation is defined by considering the space $ F _ {h} $ as normed, and by assuming, for the grid and for the norm, that for any function $ f \in F $, the function $ \{ f \} _ {h} \in F _ {h} $, which coincides with $ f $ at the points of $ D _ {hF } $, satisfies the equation

$$ \lim\limits _ {h \rightarrow 0 } \| \{ f \} _ {h} \| _ {F _ {h} } = \| f \| _ {F} . $$

The operator $ L _ {h} $ is understood to be an operator from $ U _ {h} $ in $ F _ {h} $, and one says that $ L _ {h} $ represents an order $ p $ approximation to $ L $ on $ U $ if, for $ h \rightarrow 0 $,

$$ \| \{ Lu \} _ {h} - L _ {h} [ u ] _ {h} \| _ {F _ {h} } \rightarrow 0 , $$

$$ \| \{ Lu \} _ {h} - L _ {h} [ u ] _ {h} \| _ {F _ {h} } \leq ch ^ {p} . $$

In order to construct an operator $ L _ {h} $ which is an approximation to $ L $ of given order on sufficiently smooth functions, one often replaces each derivative contained in the expression $ L $ by its finite-difference approximation, basing oneself on the following fact. For any integers $ i, j $ and for any $ k _ {0} $, $ 2 k _ {0} + 1 \geq i + j $, in the equation

$$ h ^ {-j} \sum _ {k = - k _ {0} } ^ { k } _ {0} c _ {k} u ( x + kh ) = u ^ {(j)} ( x ) + \epsilon ( x , h , c _ {-k _ {0} } \dots c _ {k _ {0} } ), $$

it is possible, by using the method of undetermined coefficients and Taylor's formula, to select numbers $ c _ {k} $ not depending on $ h $, so that for any function $ u(x) $ with $ j + r $( $ r \leq i $) bounded derivatives, an inequality of the type

$$ | \epsilon ( x , h , c _ {- k _ {0} } \dots c _ {k _ {0} } ) | \leq A _ {ij} \sup _ { t } \ | u ^ {( j + r ) } ( t ) | h ^ {r} , $$

where $ A _ {ij } $ depends only on $ i $ and $ j $, is valid. As an example, suppose one constructs an approximating operator for the Laplace operator $ \Delta $,

$$ \Delta u \equiv \frac{\partial ^ {2} u }{\partial x ^ {2} } + \frac{\partial ^ {2} u }{\partial y ^ {2} } = f ( x , y ) , $$

if $ D _ {U} $ is the closed square $ | x | \leq 1, | y | \leq 1 $, and $ D _ {F} $ is its interior $ | x | < 1, | y | < 1 $. Assume that $ h = 1/N $ where $ N $ is a natural number, and construct the grid with points

$$ ( x , y ) = ( mh, nh ) ,\ \ | mh | \leq 1,\ | nh | \leq 1, $$

which belong to $ D _ {hU } $. The points

$$ ( x , y ) = ( mh , nh ) ,\ \ | mh | \leq 1,\ | nh | \leq 1, $$

then belong to $ D _ {hF } $, for integers $ m $ and $ n $. Since

$$ \frac{1}{h ^ {2} } [ y ( x + h ) -2 y ( x ) + y ( x - h ) ] = y ^ {\prime\prime} ( x ) + h ^ \frac{2}{12} y ^ {(4)} ( \xi ) , $$

$ \Delta $ can be approximated with second-order accuracy on a space of sufficiently smooth functions by the finite-difference operator $ L _ {h} $ if one puts, at the points of $ D _ {hF } $:

$$ L _ {h} u _ {m,n} = \frac{u _ {m+1,n} -2u _ {m,n} +u _ {m-1,n} }{h ^ {2} } + $$

$$ + \frac{u _ {m,m+1} -2u _ {m,n} +u _ {m,n-1} }{h ^ {2} } = f _ {mn} , $$

where $ u _ {m,n } $ and $ f _ {m,n } $ are the values of the functions $ [u] _ {h} $ and $ f _ {h} $ at the point $ (mh, nh) $.

There are also other methods of constructing operators $ L _ {h} $ which are approximations to the operator $ L $ on the space of solutions $ u $ of the differential equation $ Lu = 0 $, 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)