Approximation of a differential equation by difference equations
An approximation of a differential equation by a system of algebraic equations for the values of the unknown functions on some grid, which is made more exact by making the parameter (mesh, step) of the grid tend to zero.
Let , be some differential operator and let
, be some finite-difference operator (cf. Approximation of a differential operator by difference operators). One says that the finite-difference expression
,
, is an approximation of order
to the differential equation
on solutions
if the operator
approximates the operator
on solutions
of order
, i.e. if
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The simplest example of the construction of a finite-difference equation which approximates the differential equation
on solutions
, consists in replacing each derivative in the expression
by its finite-difference analogue.
For instance, the equation
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is approximated with second-order accuracy by the finite-difference equation
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where the grids and
consist of points
,
is an integer, and
is the value of the function
at
. Again, the equation
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is approximated by two different difference approximations on smooth solutions:
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(explicit scheme), and
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(implicit scheme), in which the grids and
consist of the points
,
,
,
and
are integers, and
is the value of the function
at the point
of the grid. There exist finite-difference operators
which represent an approximation to the differential operator
especially well only on solutions
of the equation
and less well on other functions. For instance, the operator
,
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where , is a first-order approximation (with respect to
) of the operator
,
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on arbitrary smooth functions and it is a second-order approximation on solutions of the equation
(it is assumed that the function
is sufficiently smooth). When finding numerical solutions of boundary value problems for the differential equation
with the aid of the finite-difference equation
, it is the approximation property of
when applied to solutions
of
that is significant and not its approximation property when applied to arbitrary smooth functions. For a large class of differential equations and systems of equations there exist methods for constructing difference equations approximating them, which also meet various additional conditions: stability of the solution
with respect to rounding-off errors such as are permitted in the calculations; the validity of certain integral relations for
which hold for the solution
of the differential equation; the permissibility of using arbitrary grids
and
(this is important in calculating the motion of a continuous medium); a limitation on the number of arithmetical operations required to compute the solutions; etc.
The approximation of a differential equation by difference equations is an element of the approximation of a differential boundary value problem by difference boundary value problems in order to approximately calculate a solution of the former.
References
[1] | S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian) |
[2] | A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian) |
[3] | S.K. Godunov, et al., "Numerical solution of multi-dimensional problems of gas dynamics" , Moscow (1976) (In Russian) |
[4] | A.A. Samarskii, Yu.P. Popov, "Difference schemes of gas dynamics" , Moscow (1975) (In Russian) |
Comments
For additional references, see the additional references to Approximation of a differential operator by difference operators.
Approximation of a differential equation by difference equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_of_a_differential_equation_by_difference_equations&oldid=12012