Difference operator

An operator acting on a space of grid functions. Difference operators occur in approximating a differential-difference problem and are the subject of study in the theory of difference schemes (cf. Difference schemes, theory of). A difference scheme can be considered as an operator equation with operators acting on a certain function space, namely a space of grid functions. A space of grid functions is a set of functions defined at the points of a given grid and forming a finite-dimensional vector space. Spaces of grid functions usually occur in approximating some space of functions of a continuous variable.

Example 1. Let $ C [ 0, 1] $ be the space of continuous functions given on the interval $ 0 \leq x \leq 1 $ with norm

$$ \| u \| _ {C} = \ \max _ {x \in [ 0, 1] } \ | u ( x) | . $$

One introduces the grid

$$ \omega _ {h} = \ \{ {x _ {i} = ih } : {i = 0 \dots N; hN = 1 } \} $$

and considers the set $ C _ {h} [ 0, 1] $ of functions $ y = \{ y _ {i} \} _ {i = 0 } ^ {N} $, $ y _ {i} = y ( x _ {i} ) $, given on the grid $ \omega _ {h} $. The set $ C _ {h} [ 0, 1] $ forms an $ ( N + 1) $- dimensional vector space under coordinate-wise addition and multiplication by scalars. The norm in $ C _ {h} [ 0, 1] $,

$$ \| y \| _ {C _ {h} } = \ \max _ {0 \leq i \leq N } \ | y _ {i} | , $$

is compatible with the norm in $ C [ 0, 1] $ in the sense that for any function $ u \in C [ 0, 1] $ the vector

$$ u _ {h} = \ \{ u _ {i} \} _ {i = 0 } ^ {N} \in \ C _ {h} [ 0, 1],\ \ u _ {i} = u ( x _ {i} ), $$

is defined and

$$ \lim\limits _ {h \rightarrow 0 } \ \| u _ {h} \| _ {C _ {h} } = \ \| u \| _ {C} . $$

Any linear difference operator $ A _ {h} $, considered as an operator acting on a finite-dimensional space, can be represented by a matrix. Matrices generated by difference operators are characterized by their large size and the relatively large amount of zero elements.

In general, the construction of spaces of grid functions and difference operators can be very complicated. It is the properties of difference operators acting on spaces with a Hilbert metric which are mostly studied. In this case the most interesting properties of a difference operator are self-adjointness and positive definiteness. The difference analogues of the formulas for differentiation of a product and integration by parts are fundamental to the mathematical apparatus which makes it possible to study the properties of difference operators.

Example 2. Let a set of real-valued functions on the grid $ \omega _ {h} $ be given. One introduces the notations:

$$ y _ {\overline{x}\; , i } = \ { \frac{y _ {i} - y _ {i - 1 } }{h} } ,\ \ y _ {x,i} = \ { \frac{y _ {i + 1 } - y _ {i} }{h} } ,\ \ y _ {x,i} ^ {0} = \ \frac{y _ {i + 1 } - y _ {i - 1 } }{2h } , $$

$$ y _ {\overline{x}\; x, i } = \frac{y _ {i + 1 } - 2y _ {i} + y _ {i - 1 } }{h ^ {2} } ,\ ( y, v) = \ \sum _ {i = 1 } ^ { {N } - 1 } y _ {i} v _ {i} h, $$

$$ ( y, v] = \sum _ {i = 1 } ^ { N } y _ {i} v _ {i} h,\ [ y, v) = \sum _ {i = 0 } ^ { {N } - 1 } y _ {i} v _ {i} h. $$

The following formulas are valid:

$$ ( yv) _ {\overline{x}\; , i } = \ y _ {\overline{x}\; , i } v _ {i} + y _ {i} v _ {\overline{x}\; , i } - hy _ {\overline{x}\; , i } v _ {\overline{x}\; , i } , $$

$$ ( yv) _ {x,i} = y _ {x,i} v _ {i} + y _ {i} v _ {x,i} + hy _ {x,i} v _ {x,i} . $$

One also has the formula for summation by parts:

$$ ( y, v _ {\overline{x}\; } ] = \ - [ y _ {x} , v) + y _ {N} v _ {N} - y _ {0} v _ {0} . $$

The latter formula implies, in particular, that the second-difference derivative operator

$$ ( Ay) _ {i} = \ - y _ {\overline{x}\; x, i } ,\ \ i = 1 \dots N - 1, $$

is self-adjoint and positive definite on the set of functions defined on $ \omega _ {h} $ and vanishing on the boundary $ i = 0 $, $ i = N $.

Much research has been devoted to the study of properties of difference approximations to differential operators of elliptic type (see [1]–[4]). To construct the corresponding difference operators effective methods can be used, such as the balance method, the finite-element method, variation and projection methods. The difference approximations that have been obtained are good models of the basic properties of the original operators such as, for example, ellipticity, fulfilling the maximum principle, etc. Difference operators have also been constructed for approximating elliptic differential operators in complicated types of regions with various kinds of boundary conditions and also for irregular grids (see [5]).

The properties of stationary difference operators are used to study the stability of non-stationary difference problems and to construct iteration methods. Moreover, the theory of iteration methods can be presented as one of the parts of the general theory of stability for difference schemes (see [6], [7]).

Factorized difference operators have been studied in connection with the construction of economic difference schemes for multi-dimensional problems in mathematical physics. These are multi-dimensional difference operators that can be represented as a product of one-dimensional difference operators (see [1]). Non-linear difference operators have also been studied (see ).

References

[1]	A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)
[2]	A.A. Samarskii, V.B. Andreev, "Méthodes aux différences pour équations elliptiques" , MIR (1978) (Translated from Russian)
[3]	V.G. Korneev, "Schemes for the finite element method with a high order of accuracy" , Leningrad (1977) (In Russian)
[4]	J.-P. Aubin, "Approximation of elliptic boundary-value problems" , Wiley (1972)
[5]	A.A. Samarskii, I.V. Fryazinov, "Difference approximation methods for problems of mathematical physics" Russian Math. Surveys , 31 : 6 (1976) pp. 179–213 Uspekhi Mat. Nauk , 31 : 6 (1976) pp. 167–197
[6]	A.A. Samarskii, A.V. Gulin, "Stability of difference schemes" , Moscow (1973) (In Russian)
[7]	A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian)
[8a]	M.M. Karchevskii, A.D. Lyashko, "Difference schemes for nonlinear multidimensional elliptic equations I" Izv. Vyzov. Mat. , 11 (1972) pp. 23–31 (In Russian)
[8b]	M.M. Karchevskii, A.D. Lyashko, "Difference schemes for nonlinear multidimensional elliptic equations II" Izv. Vyzov. Mat. , 3 (1973) pp. 44–52 (In Russian)

Comments

For references see also Difference equation and Difference scheme.

References

[a1]	S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)