View source for Closure of a computational algorithm

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A system of equations

$$L_zu=f_z,\quad0\leq z\leq Z,\label{1}\tag{1}$$

obtained as the limit as $h\to0$, $z(m,h)\to z$ of a system of partially solved equations

$$L_m^hu^h=f_m^h,\quad m=0,\dots,M,\label{2}\tag{2}$$

describing the successive steps of a computational algorithm for the solution of an equation

$$L^hu^h=f^h\label{3}\tag{3}$$

(for example, a finite-difference equation, in which case $h$ is the grid spacing), which approximates the equation

$$Lu=f\label{4}\tag{4}$$

as $h\to0$. It is assumed here that $L_0^h=L^h$, $f_0^h=f^h$, $L_M^h$ is the identity operator, and that $f_M^h=(L^h)^{-1}f^h=u^h$, i.e. the $M$-th step of the algorithm yields a final solution of the approximate equation \eqref{3}. The function $z(m,h)$ is assumed to increase with $m$ (e.g. is a linearly increasing function) and to satisfy the boundary conditions $z(0,h)=0$, $z(M,h)=Z$. The case $M=\infty$ is admissible; then $L_\infty^h$, $f_\infty^h$, $z(\infty,h)$ are interpreted as the limits of the variables $L_m^h$, $f_m^h$, $z(m,h)$ as $m\to\infty$. The case $M=\infty$ corresponds to iteration methods for solving equation \eqref{3}.

If the operators $L_z$ in equation \eqref{1} are uniformly bounded in $z$, one says that algorithm \eqref{2} admits a regular closure. Although the set of algorithms with regular closure does not coincide with the set of actually stable algorithms, construction of a closed algorithm frequently helps in investigating the stability of an algorithm under various perturbations (in particular computational errors) (see [[#References|[3]]], [[#References|[4]]]).

The concept of a closed algorithm was introduced in [[#References|[1]]]. There the closure of an algorithm for successive elimination of unknowns, when solving finite-difference equations that approximate an equation \eqref{4} with $Lu=u-Au$, where $A$ is a Fredholm integral operator, was obtained and investigated.

The construction of the closure of an algorithm and the inverse operation — the construction of a discrete algorithm the closure of which is a given continuous process — proves useful in designing new methods for the solution of problems. In particular, a large number of iteration methods admit closures which are steady processes. For example, the simple iteration method for the solution of the Laplace difference equation corresponds to the steady process $u_t=\Delta u$; and the two-step iteration method corresponds to the steady process $u_{tt}+\alpha u_t=\Delta u$, $\alpha>0$ (see [[#References|[5]]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  S.L. Sobolev,   "Some remarks on the numerical solution of integral equations"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''20''' :  4  (1956)  pp. 413–436  (In Russian).  Zbl 0074.11004.</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  I. [I. Babushka] Babuška,   M. Práger,   E. Vitásek,   "Closure of computational processes and the double-sweep method"  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''4''' :  2  (1964)  pp. 351–353  (In Russian)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  N.S. Bakhalov,   "Computational methods for the solution of ordinary differential equations" , Kiev  (1970)  (In Russian)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  V.K. Saul'ev,   "Integration of equations of parabolic type by the method of nets" , Pergamon  (1964)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  A.F. Shapkin,   "Closure of two computational algorithms, based on the idea of orthogonalization"  ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''7''' :  2  (1967)  pp. 411–416  (In Russian)</TD></TR>
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====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. [I. Babushka] Babuška,   M. Práger,   E. Vitásek,   "Numerical processes in differential equations" , Wiley  (1966).  Zbl 0156.16003</TD></TR></table>