Geodesic
Discretization of second-order ordinary differential equations with symmetries
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1329-1355

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A number of publications (indicated in the Introduction) are overviewed that address the group properties, first integrals, and integrability of difference equations and meshes approximating second-order ordinary differential equations with symmetries. A new example of such equations is discussed in the overview. Additionally, it is shown that the parametric families of invariant difference schemes include exact schemes, i.e., schemes whose general solution coincides with the corresponding solution set of the differential equations at mesh nodes, which can be of arbitrary density. Thereby, it is shown that there is a kind of mathematical dualism for the problems under study: for a given physical process, there are two mathematical models: continuous and discrete. The former is described by continuous curves, while the latter, by points on these curves. 
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V. A. Dorodnitsyn; E. I. Kaptsov. Discretization of second-order ordinary differential equations with symmetries. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1329-1355. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a9/