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
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
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  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 , –, 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
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
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 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 .
The replacement of differential problems by their difference analogues started in the works ,  and , 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 .
|||L.A. Lyusternik, "Dirichlet's problem" Uspekhi Mat. Nauk , 8 (1940) pp. 125–124 (In Russian)|
|||R. Courant, K. Friedrichs, H. Lewy, "Ueber die partiellen Differenzengleichungen der mathematischen Physik" Math. Ann. , 100 (1928)|
|||S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)|
|||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)|
|||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)|
|||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)|
|||A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)|
|||A.F. Filippov, "On stability of difference equations" Dokl. Akad. Nauk SSSR , 100 : 6 (1955) pp. 1045–1048 (In Russian)|
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. V.S. Ryaben'kii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Approximation_of_a_differential_boundary_value_problem_by_difference_boundary_value_problems&oldid=13955