Approximation of a differential boundary value problem by difference boundary value problems
An approximation of a differential equation and its boundary conditions by a system of finite (usually algebraic) equations giving the values of the unknown function on some grid, which is subsequently made more exact by making the parameter of the finite-difference problem (the step of the grid, the mesh) tend to zero.
Consider the computation of a function which belongs to a linear normed space
of functions defined in a given domain
with boundary
, and which is the solution of the differential boundary value problem
,
, where
is a differential equation, while
is the set of boundary conditions. Let
be a grid (cf. Approximation of a differential operator by difference operators) and let
be the normed linear space of functions
defined on this grid. Let
be a table of values of the function
at the points of
. A norm is introduced into
so that the equality
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is valid for any function . The problem of computing the solution
is replaced by a certain problem
for the approximate computation of the table
of values of
at the points of
. Here,
is a certain set of (non-differential) equations for the values of the grid function
.
Let be an arbitrary function of
, let
, and let
be the normed linear space to which
belongs for any
. One says that the problem
is a finite-difference approximation of order
of the differential boundary value problem
,
, on the space of solutions
of the latter if
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The actual construction of the system involves a separate construction of its two subsystems
and
. For
one uses the difference approximations of a differential equation (cf. Approximation of a differential equation by difference equations). The complementary equations
are constructed using the boundary conditions
.
An approximation such as has just been described never ensures [2] that the solution of the finite-difference problem converges to the exact solution
, i.e. that the equality
![]() |
is valid, no matter how the norms in and
have been chosen.
The additional condition, the fulfillment of which in fact ensures convergence, is stability [3], [5]–[8], which must be displayed by the finite-difference problem . This problem is called stable if there exist numbers
and
such that the equation
has a unique solution
for any
,
,
, and if this solution satisfies the inequality
![]() |
where is a constant not depending on
or on the perturbation
of the right-hand side, while
is a solution of the unperturbed problem
. If a solution
of the differential problem exists, while the finite-difference problem
approximates the differential problem on solutions
of order
and is stable, then one has convergence of the same order, i.e.
![]() |
For instance, the problem
![]() | (1) |
where is a given function with a bounded second-order derivative, can be approximated, for a natural definition of the norms, by the finite-difference problem
![]() | (2) |
![]() |
where is the value of
at
,
,
. If the norm of
is taken to be the upper bound of the moduli of the right-hand sides of the equations which constitute the system
,
, then the approximation of problem (1) by problem (2) on solutions
is of the first order. If
, there is no convergence, whatever the norm. If
and the norm is
![]() |
the problem is stable, so that there is convergence [2], [3]:
![]() |
The replacement of differential problems by difference problems is one of the most universal methods for the approximate computation of solutions of differential boundary value problems on a computer [7].
The replacement of differential problems by their difference analogues started in the works [1], [2] and [4], and is sometimes employed to prove that the differential problem is in fact solvable. This is done as follows. It is proved that the set of solutions of the difference analogue of the differential boundary value problem is compact with respect to
, after which a proof is given that a solution
of the differential boundary value problem is the limit of a subsequence
which converges as
. If this solution is known to be unique, then not only the subsequence, but also the entire set of
converges to the solution
as
.
References
[1] | L.A. Lyusternik, "Dirichlet's problem" Uspekhi Mat. Nauk , 8 (1940) pp. 125–124 (In Russian) |
[2] | R. Courant, K. Friedrichs, H. Lewy, "Ueber die partiellen Differenzengleichungen der mathematischen Physik" Math. Ann. , 100 (1928) |
[3] | S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian) |
[4] | I.G. Petrovskii, "New existence proofs for the solution of the Dirichlet problem by the method of finite differences" Uspekhi Mat. Nauk , 8 (1940) pp. 161–170 (In Russian) |
[5] | V.S. Ryaben'kii, "On the application of the method of finite differences to the solution of the Cauchy problem" Dokl. Akad. Nauk SSSR , 86 : 6 (1952) pp. 1071–1073 (In Russian) |
[6] | V.S. [V.S. Ryaben'kii] Rjabenki, A.F. [A.F. Filippov] Filipov, "Über die stabilität von Differenzgleichungen" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian) |
[7] | A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian) |
[8] | A.F. Filippov, "On stability of difference equations" Dokl. Akad. Nauk SSSR , 100 : 6 (1955) pp. 1045–1048 (In Russian) |
Comments
For additional references, see the additional references to Approximation of a differential operator by difference operators.
Approximation of a differential boundary value problem by difference boundary value problems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_of_a_differential_boundary_value_problem_by_difference_boundary_value_problems&oldid=13955