Continuous versus discrete structures I -- Discrete embeddings and   ordinary differential equations
  We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit three main discrete associate : the differential, integral or variational discrete embeddings which corresponds to classical numerical scheme including variational integrators. 
Jacky Cresson and Fr\'ed\'eric Pierret
math.NA math.CT