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A method in which the solution  $  u $
of a stationary problem

$$ \tag{1 }
A u  =  f
$$

is regarded as the steady-state limit solution for  $  t \rightarrow \infty $
of a Cauchy initial value problem for a non-stationary evolution equation involving the same operator  $  A $ (cf. [[Cauchy problem|Cauchy problem]]). This evolution equation may e.g. be of the form

$$ \tag{2 }
\sum _ { i=1 } ^ { m } 
C _ {i} 
\frac{d  ^ {i} u (t) }{d t  ^ {i} }
  = \ 
f - A u (t) ,
$$

$$ 
\left . 
\frac{d  ^ {k} u }{d t  ^ {k} }
 \right | _ {t=0}  =  u _ {0k} ,\  k = 0 \dots m - 1 .
$$

Here the  $  C _ {i} $
are suitable operators which guarantee the existence of the  "adjustment limit"   $  \lim\limits _ {t \rightarrow \infty }   u (t) = u $.

A result of using adjustment is that it permits one to use approximate solution methods of (2) in order to construct iteration algorithms for solving equation (1) (cf. [[Iteration algorithm|Iteration algorithm]]). Thus, for the non-stationary equation (2) one could employ a discretization (differencing) with respect to  $  t $
solution method which is convergent and stable to obtain approximate solutions. For example, for  $  m = 1 $, 
an explicit method of the form

$$ 
C _ {1} 
\frac{u ( t _ {n+1} ) - u ( t _ {n} ) }{\tau _ {n} }
  = \ 
f - A u ( t _ {n} )
$$

where  $  \tau _ {n} = t _ {n+3} - t _ {n} > 0 $. 
And then this method can be interpreted as an iteration algorithm

$$ 
C _ {1} ( u  ^ {n+1} - u  ^ {n} )  = \ 
\tau _ {n} ( f - A u  ^ {n} ) ,\ \ 
n = 0 , 1 \dots \ \ 
u  ^ {0}  =  u _ {00} ,
$$

for solving equation (1), in which  $  C _ {1} $
and  $  \tau _ {n} $
are now seen as characterizing this (iteration) method.

Varying the form of the operators  $  C _ {i} $
and considering different discretizations with respect to  $  t $
in equation (2) (explicit schemes, implicit schemes, splitting schemes, etc.) gives the possibility of obtaining a wide variety of iteration methods for solving equation (1). For these methods equation (2) will be the closure of the computational algorithm (cf. [[Closure of a computational algorithm|Closure of a computational algorithm]]). A generalization of the adjustment method is the [[Continuation method (to a parametrized family)|continuation method (to a parametrized family)]].

====References====
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  S.K. Godunov,   V.S. Ryaben'kii,   "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.I. Marchuk,   V.I. Lebedev,   "Numerical methods in the theory of neutron transport" , Harwood  (1986)  (Translated from Russian)</TD></TR></table>

====Comments====
The adjustment method is also called the time-stepping method. Such a time-stepping method with  $  m = 1 $
arises quite naturally when an elliptic boundary value problems is viewed as a steady state of a (dissipative) parabolic problem (cf. [[#References|[a2]]]). Because of the inherent (numerical) stiffness an implicit discretization method, such as BDF, should be advocated.

Similar ideas can be used in optimization theory by e.g. constructing a mechanical system whose stable equilibrium state is the desired optimum. Again this leads to all kinds of iteration algorithms, as in [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Babushka,   S.L. Sobolev,   "The optimization of numerical processes"  ''Appl. Mat.'' , '''10'''  (1965)  pp. 96–130</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Kubicek,   V. Hlavacek,   "Numerical solution of nonlinear boundary value problems with applications" , Prentice-Hall  (1983)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B.S. Razumikhin,   "Physical models and equilibrium methods in programming and economics" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.C. Rheinboldt,   "Numerical analysis of parametrized nonlinear equations" , Wiley  (1986)</TD></TR></table>