Model for calculations

computational model

A typical problem used as a model for investigating and developing numerical methods for some class of problems. For example, in the theory of quadrature the problem of calculating integrals of functions satisfying a condition $ | f ^ { ( n) } | \leq A $ is considered. The processing of methods for the solution of the Cauchy problem for systems of ordinary differential equations historically was done by investigating the properties of the methods on models from a sequence of increasing complexity (with integration interval $ [ 0 , X ] $):

1) the equation $ y ^ \prime = 0 $;

2) the equation $ y ^ \prime = m y $, $ | m | X $ of order 1 (models 1) and 2) correspond to the problem of integration on small time intervals of systems with smooth solutions);

3a) the equation $ y ^ \prime = m y $, $ m < 0 $, $ | m | X \gg 1 $; this model corresponds to the problem of integration on large time intervals of systems with stable solutions;

3b) the equation $ y ^ \prime = x ^ \lambda $; a model of an equation with singularities in the derivatives of solutions;

4) the system $ y _ {1} ^ \prime = m _ {1} y _ {1} $, $ y _ {2} ^ \prime = m _ {2} y _ {2} $, $ 0 > m _ {1} > m _ {2} $, $ | m _ {2} | X \gg | m _ {1} | X $, $ | m _ {1} | X $ of order 1; a model of so-called stiff differential systems (cf. Stiff differential system), in which one component varies relatively slowly and the other rapidly.

References

[1]	N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) MR0362811 Zbl 0524.65001