Various closeness concepts in numerical ODEʹs

We present several results pointing out that appropriate concepts of closeness between exact solutions and their discretization approximation depend on the inner structure of the underlying differential equations. In many cases, especially if function τ is not bounded, the exact solution segment {ф(t, x) ε Rm t [0, τ(x))} is suggested to be compared witha (h,•)-invariant (i.e., overflowing-invariant with respect to the one-step discretization mapping (h,•)) curve starting at some point J(x) Rm. Properties like continuity, surjectivity, bijectivity, and nearness-to-identity of the assignment x → Jh(x) are also considered.