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Several new methods of numerical integration of Cauchy problems with blow-upsolutions for nonlinear ordinary differential equations of the first- andsecond-order are described. Solutions of such problems have singularities whosepositions are unknown a priori (the standard numerical methods for solvingproblems with blow-up solutions can lead to significant errors). The firstproposed method is based on the transition to an equivalent system of equationsby introducing a new independent variable chosen as the first derivative. Thesecond method is based on introducing a new auxiliary non-local variable withthe subsequent transformation to the Cauchy problem for the correspondingsystem of ODEs. The third method is based on adding to the original equation ofa differential constraint, which is an auxiliary ODE connecting the givenvariables and a new variable. The proposed methods lead to problems whosesolutions are represented in parametric form and do not have blowing-upsingular points; therefore the transformed problems admit the application ofstandard fixed-step numerical methods. The efficiency of these methods isillustrated by solving a number of test problems that admit an exact analyticalsolution. It is shown that: (i) the methods based on non-local transformationsof a special kind are more efficient than several other methods, (ii) among theproposed methods, the most general method is the method based on thedifferential constraints. Some examples of nonclassical blow-up problems areconsidered. Simple theoretical estimates are derived for the critical value ofan independent variable. It is shown that the method based on a non-localtransformation of the general form as well as the method based on thedifferential constraints admit generalizations to the $n$th-order ODEs andsystems of coupled ODEs.

Numerical integration of blow-up problems on the basis of non-local transformations and differential constraints