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A method in which the solution of a stationary problem

 (1)

is regarded as the steady-state limit solution for of a Cauchy initial value problem for a non-stationary evolution equation involving the same operator (cf. Cauchy problem). This evolution equation may e.g. be of the form

 (2)

Here the are suitable operators which guarantee the existence of the "adjustment limit" .

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). Thus, for the non-stationary equation (2) one could employ a discretization (differencing) with respect to solution method which is convergent and stable to obtain approximate solutions. For example, for , an explicit method of the form

where . And then this method can be interpreted as an iteration algorithm

for solving equation (1), in which and are now seen as characterizing this (iteration) method.

Varying the form of the operators and considering different discretizations with respect to 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). A generalization of the adjustment method is the continuation method (to a parametrized family).

#### References

 [1] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) [2] S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian) [3] G.I. Marchuk, V.I. Lebedev, "Numerical methods in the theory of neutron transport" , Harwood (1986) (Translated from Russian)